How do Catholics and other Christians respond to the US Bishops' teaching that Matthew was not written by it supposed author?

There are 3 errors in this statement

USCCB has absolutely no teaching authority on this question

It is not a matter of doctrine one way or the other and to put it forward as dogmatic would be heretical

From Pope Benedict's audience in 2006 on Matthew: "The Gospel of Matthew written in Hebrew or Aramaic is no longer extant, but in the Greek Gospel that we possess we still continue to hear, in a certain way, the persuasive voice of the publican Matthew, who, having become an Apostle, continues to proclaim God's saving mercy to us. And let us listen to St Matthew's message, meditating upon it ever anew also to learn to stand up and follow Jesus with determination."

Markian priority is bunk. The USCCB can make an error like this because they are trusting poor scholarship and publishing an edition of the bible based on poor scholarship on their website. That edition based on poor scholarship contains an error written by an uninspired translator or commentator who got things wrong.

Hold to the Rod's answer is good, but he misses a third possibility: neither Matthew nor Mark quote one another. Father Sebastian Carnazzo, a Melkite Catholic priest (Melkites are an Eastern Rite in full communion with Rome) has a video series on the Institute of Catholic Culture website (this is free and accessible to the public) which constitutes a New Testament bible study. He explains the so-called Markian "problem" as a non-problem. According to him, the ancient Church was reciting the gospel story in their liturgies from memory, and these records pulled from that liturgical practice. It therefore follows that one gospel author needn't quote directly from another. Both can be pulling from the oral tradition - and likely are.

Matthew probably wrote Matthew. That's why the early Church named the gospel account after him. There probably was not a Q source, unless one wants to consider the oral tradition Q.

The Pontifical Biblical Commission under Pope St. Pius X already settled the dubium, on 19 June 1911, regarding St. Matthew's authorship of the first gospel:

1. Seeing the universal and constant tradition of the Church dating from the first centuries, which explicit testimonies of the Fathers, the inscriptions of the codices of the Gospels, the oldest version of the sacred books as well as their catalogues transmitted to us by the holy Fathers, ecclesiastical writers, Supreme Pontiffs and the Councils, and finally, the liturgical usages of the Eastern and Western Church clearly record, it may and must be affirmed with certainty that Matthew, an Apostle of Christ, is in truth the author of the Gospel published under his name.

Answer: Yes.

Acta Apost. Sedis 3 (1911) p. 294; also in: Enchiridion Biblicum §401 (p. 112) of the Latin original or §383 of the English transl. and Denzinger 2148

I made a series of videos on this very question! I'm not a Catholic, but I am a Christian, so I think that leaves me in scope for this question. My work is here: Who When & Why - the Writing of the Gospels.

Among other things, it:

• Traces the belief that Matthew wrote Matthew to the first century
• Makes the case that the hypothesis of Markan Priority (Mark's Gospel was written first) is at best reliant on circular reasoning, and at worst, untenable
• Argues that Matthean Priority (Matthew's Gospel was written first) is the best explanation of the evidence.

In response to the OP's specific questions, I'll offer an answer 2 ways.

First I'll evaluate the question as stated. Second, I'll offer an alternate explanation of the evidence. Both approaches will offer an answer to the question, though in very different ways.

1. Matthew quoted Mark

Papias of Hieropolis preserved in his history (written ~105) the testimony of a first-generation Christian Elder, indicating that Mark wrote a Gospel based on the preaching of Peter. Clement of Alexandria recorded this information a century later as well. Since Clement provides a number of details not found in any known fragment of Papias (and since Clement was from the Alexandrian church Mark is said to have founded), it is likely that Clement has at least some independent information (For the testimony of Papias, see HE 3.39; for several of the key statements by Clement see HE 2.15 & HE 6.14).

If the Gospel of Mark was based on Peter's preaching, and Matthew considered it reliable and reasonably well-stated, why reinvent the wheel? Peter would seem an excellent source from which Matthew could derive a portion of his material.

Matthew, writing to a Jewish audience, would have rewritten some of what Mark wrote (to Romans), but the core of Mark's Gospel would work just fine as the foundation from which to start.

To use a modern example, it is not unheard of for an eyewitness to an event to quote a statement about the event that was recorded by a journalist. If the journalist did good research and said it well--even if the journalist was not an eyewitness--why not quote their work?

2. Matthew did not quote Mark

I say this knowing full well I'm fighting against the current. Markan Priority (the view that Mark was the first of the Synoptic Gospels written) has been very popular in New Testament scholarship since the 1870s in German and the 1920s in English. I offer a critique of the major arguments for Markan Priority on Hermeneutics here, and an extensive critique on my YouTube channel here.

I do not believe that Matthew's Gospel is based on Mark's, but that Mark's Gospel was based on Matthew's. If Mark did not precede Matthew, the answer to the OP's question would be a simple: he didn't.

Frank Luke offers an excellent summary of the case for Matthean Priority (that Matthew was the first of the Synoptic Gospels) on Hermeneutics here.

My principal arguments for Matthean Priority are:

1. The Argument from Order -- Mark's ordering of the material is sensible if Mark wrote 3rd. Matthew or Luke's order is implausible if either of them wrote third. My thoughts here.
2. The Patristic Testimony -- the early Christian historians are unanimous in their testimony that Matthew wrote first. My thoughts here.
3. The fact that the Jewish Gospel of Matthew was the most popular Gospel in the early Gentile church. This is sensible if Matthew was considered the primary source. This is difficult to rationalize if the Gospel that calls Gentiles "dogs" upstaged a Gentile Gospel (Mark or Luke) and was written by a Jewish Christian during the time period in which Judaism & Christianity went through their final, painful separation. My thoughts here.

Conclusion

Why did Matthew quote Mark? I believe it is possible to argue for why an eyewitness would use another's work if he considered the work well-done. I believe it is even easier to answer the question by saying Matthew didn't quote Mark.

I will add to what has already been stated. It is interesting how the modern Bishops (influenced by hyper Protestant skepticism?) object to what the Pontifical Biblical Commission under Pope St. Pius X wrote back in 1911:

Seeing the universal and constant tradition of the Church dating from the first centuries, which explicit testimonies of the Fathers, the inscriptions of the codices of the Gospels,.. finally, the liturgical usages of the Eastern and Western Church clearly record.

In support of Pope Pius X and the old Pontifical Biblical Commission, there is an early reliable church tradition attributed to Clement of Alexandria that the Gospels with the genealogies (Matthew and Luke) were “written first” (progegraphthai). See Eusebius' Church History, 6:14:6-7.

However, Stephen Carlson argues that it possible that the key Greek verb (progegraphthai) should be rendered “published openly” rather than “written first.” If so, Clement was claiming that Matthew and Luke were published openly, while Mark was initially written for a group of private individuals, without Peter’s initial knowledge or authorization.

According to Irenaeus, in his work, Against Heresies (3.1.1), Matthew issued a written Gospel among the Hebrews in their own dialect, while Peter and Paul were preaching at Rome. After their departure, Mark, the disciple and interpreter of Peter, handed down in writing what had been preached by Peter. So it appears that it is while Peter and Paul were in Rome that Matthew wrote, but it was after they departed that Mark wrote. Mark and Peter may not have had Matthew's original Gospel (written in Jerusalem?) to use, or even to endorse.

In his De Viris Illustribus, Jerome states that Matthew wrote in Hebrew letters and words for the sake of the Jews and then it was translated into Greek. He writes that "the Hebrew itself is preserved even now in the library at Caesarea...” Although, to be sure, later on Jerome wrote how the Hebrew original texts were nowhere extent. See here. So, it could be that the version Jerome spoke of was one of the many expanded interpretations of Matthew's original work that Papias spoke of and that Luke alludes to in his opening of his Gospel.

Jerome also says he "was given the opportunity of transcribing this volume by the Nazarenes who use it in Beroea, a city of Syria.” He adds that Matthew, when quoting from the Old Testament, had used the Hebrew Scriptures not the Greek Septuagint (RO 203 & DVI, ch 3&7). See here. This is interesting in that the Greek version of Matthew's Gospel often quotes from a Greek translation of the Old Testament, the Septuagint.

That reference from Jerome appears to indicate that Matthew's original Hebrew Gospel was given a "high level" translation into Greek.

Daniel Wallace writes the following:

... Matthew might have written several pamphlets of dominical sayings in Aramaic. This is what Papias is referring to (λογία, after all, is not “acts” but “discourses, sayings,”). When Mark’s Gospel was published, Matthew’s audience wanted a framework for the sayings of Jesus. It would have been at this time that Matthew organized the sayings into five thematic units, and used Mark’s Gospel as a framework for them. One of the evidences of this internally is that the narrative material in Matthew is almost merely “stage setting” for the didactic material—each narrative section (except for the birth and passion narratives) concludes with a message by Jesus. The point is that Matthew himself may well have written a document very much like Q (is it even possible that he wrote Q?!).

That being said, it should be pointed out that scholars note that the second temple Jews, including Philo, used the word λογία to refer to scripture.  

Wallace suggests the possibility that Mark recorded Peter's sermons:

... while Peter was still alive. At about the same time, Matthew published isolated sayings of Jesus in Aramaic for his and other Jewish-Christian communities. He would, therefore, have been unaware of Mark’s work, just as Mark would have been unaware of Matthew’s. Over the next few years, the dominical material of Matthew would have been translated into Greek. At the same time, Matthew’s own community wanted a framework for these sayings, in light of the publication of Mark’s Gospel. Mark was at hand for the framework, and some of Mark’s material duplicated Matthew’s (e.g., the Olivet Discourse) and was already in Greek. Hence, Matthew used Mark as his basic framework, even where sermonic material was found in Mark. Then, Matthew reorganized these isolated sayings of Jesus into five great sermons (though one was already found in Mark—viz., the Olivet Discourse). For the rest, Matthew simply supplemented Mark with a fulfillment-motif, birth narrative, etc. This hypothesis both affirms Markan priority and Papias’ statement about Matthew’s ‘Hebrew.’ As well, it strongly affirms that Matthew implicitly recognized the reliability of Mark’s Gospel.

Perhaps for the high level translation, Matthew used parts of Mark's Greek version and then reorganized his original work on the sayings of Jesus into sermonic sections?

The Bishops write:

... it is hardly likely that a companion of Jesus would have followed so extensively an account that came from one who admittedly never had such an association rather than rely on his own memories.

Actually, considering some of the divisions going on in the early Christian community, it is likely that a companion of Jesus like Matthew would have wanted to endorse the preaching of Peter that was written by Mark. A subtle way to do that would be to utilize Mark's Gospel as reference points in highlighting certain aspects of the life of Jesus for catechetical purposes.

In short, Matthew might have supplemented Mark by translating his Hebrew/Aramaic Gospel (Papias, Εβραιδι διαλεκτω) by utilizing Mark's Gospel as reference points.

At any rate, there is no need to dispute the early church tradition. In fact, using the "Ancient Documents" principle in the legal field, the benefit of the doubt should be given to the claims of those who were custodians of the original manuscripts. For example, in McCormick's Handbook of the Law of Evidence (1972: 549, Section 223 and 747, Section 323), there is this summation:

An ancient writing is usually regarded as sufficiently authenticated if the offering party shows that it has come from high antiquity and is unsuspicious in appearance [no evident marks of forgery], if found in a place of custody natural for such writing [found in the proper repository], and deemed by the law to be authentic and credible. The age requirement probably assures that there will be a special need for dispensing with the heresy rule . . . In such cases [where ancient writings are sufficiently authenticated], the burden of proof to the contrary devolves upon the objector.

The Pontifical Biblical Commission's argument for Matthew being the Gospel writer of the book in his name can be sufficiently sustained by the explicit testimonies of the church fathers. They were the custodians of the manuscripts, along with the inscriptions contained in the codices of the Gospels. The formulations εὐαγγέλιον κατά + name or κατά + name are the same in all the Gospels. See Simon Gathercole's The Titles of the Gospels in the Earliest New Testament Manuscripts.

The titles of the Gospels appear to indicate that the evangelists are not meant to appear as "biographical" authors, but to bear witness in their works to the one saving message of Jesus Christ.

The skeptical argument is that the titles operate more as placeholder names, where the Gospels have been “handed down” by church traditions affixed to names of figures in the early church, rather than the author being clearly identified.

However, just like the Gospels of Mark and Luke were written by them and not just the sources of them are from their teachings, so it is likely to assume that Matthew was the author of the Gospel of Matthew. This is also corroborated by the unanimous claims of the earliest church fathers that write about the origin of the Gospels.