Why does the Trinitarian Formula start with "In the NAME…" and not "In the NAMES…"?

It is a rhetorical device, rather than grammatical. Try reciting the formulation aloud, along with your alternative, and see which one sounds better. IMHO the "Name" formula has more of a ring to it.

The fact that it is a singular name thrice, means that it is not one baptism but 3 baptism you take in the name of each: Father, Son, Holy Spirit. Ellipsis was very common in ancient writings, and it must have been used the same way in verbal teaching or tradition in the early church, as the construct would be unnecessary to be spoken in minute detail: (baptise in the name of the father, in the name of the Son etc). Thus, you baptise thrice in the water for each name.

It could be said (either sarcastically or seriously) that the one baptism expresses a unification of the three into one name and one person, hence has been committing a Unitarian heresy. In the Catholic Church, even the use of wrong pronoun "we" instead of "I", nullifies the baptism ritual.

To some it may seem that the English translation is imitating the grammatical discord or irregular Greek pattern of collective singular nouns or verbs in this for singular subject. However, some have suggested that the ellipsis is apparent, and the sentence is legal according to English rules. The repetition of in the name of has been elided.

I found a great answer on BHSE:

Grammatically it is natural to view the prepositional phrase “in the name” as being “applied” to all three terms independently.

This could be an example of the adjunctive και[and] or of ellipsis, or both. The adjunctive και [and] lends itself to ellipsis and also serves to reduce repetition.

This would result in:

In the name of [εἰς τὸ ὄνομα] the Father and [in the name of] the Son and [in the name of] the Holy Spirit. (NA28) εἰς τὸ ὄνομα τοῦ πατρὸς καὶ τοῦ υἱοῦ καὶ τοῦ ἁγίου πνεύματος,

An even better source confirming the ellipsis is from the 19th century scholar Heinrich Meyer's NT commentary, in the footnote we read:

Had Jesus used the words τὰ ὀνόματα [names] instead of τὸ ὄνομα [name], then, however much He may have intended the names of three distinct persons to be understood, He would still have been liable to be misapprehended, for it might have been supposed that the plural was meant to refer to the various names of each separate person. The singular points to the specific name assigned in the text to each of the three respectively, so that εἰς τὸ ὄνομα is, of course, to be understood both before τοῦ υἱοῦ and τοῦ ἁυίου πνεύματος; comp. Rev 14:1 : τὸ ὄνομα αὐτοῦ καὶ τὸ ὄνομα τοῦ πατρὸς αὐτοῦ. We must beware of making any such dogmatic use of the singular as to employ it as an argument either for (Basilides, Jerome, Theophylact) or against (the Sabellians) the orthodox doctrine of the Trinity.

The example can be confused with the grammatical discord found in Greek, Beginning with NT Greek where we often see singular noun or verbs for plural subjects.

To use the example of 1 John 4:1 "Beloved, believe not every spirit, but prove the spirits, whether they are of God: because many false prophets are gone out into the world." The Greek has a singular verb "is" (estin) for plural spirits, implying implicitly to test each spirit individually. We also see the discord in Matt 10:2 (ὀνόματά ἐστιν) names is; Luke 10:20 names written(sg). However, in the case of the baptismal formula verse, it is not an example of grammatical discord, but of ellipsis.

Although one answer says that the English of grammar requires plural names for plural nouns, there are better experts like Meyer who have settled any doubt among those with poor English. We should conclude that the ellipsis is natural in English as it is in Greek. The verse should be understood as baptising in all the three names.

To be honest, I suspect this is a question of English grammar rather than theology.

The expression could be taken as an abbreviation of "In the name of the Father and {in the name] of the Son and [in the name] of the Holy Spirit". In other words, the word "name" at the beginning would be singular in form even if they were different names.

I can't speak for Latin grammar, but the point is worth checking.

"In the name…" (singular) shows the unity of the divine essence.

Pohle-Preuss, The Divine Trinity: A Dogmatic Treatise, § "The threefold personality of God as taught in the New Testament ­— Texts treating of the three Divine Persons together":

The essential identity of the three Divine Persons follows further from the singular form “in nomine” ["in the name"], because throughout the Bible “nomen Domini” ["name of the Lord"] signifies God’s power, majesty, and essence.¹³ As the Three have but one name, so They have but one essence, one nature, one substance. St. Augustine beautifully observes:

^{Iste unus Deus, quia non in nominibus Patris et Filii et Spiritus Sancti, sed in nomine Patris et Filii et Spiritus Sancti. Ubi unum nomen audis, unus est Deus}
This is one God, for it is not in the names of the Father, and of the Son, and of the Holy Ghost, but in the name of the Father, and of the Son, and of the Holy Ghost. Where thou hearest one name, there is one God.¹⁴

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<sup>13. “Nomen est numen.” ["The name is the divinity."]
14. August., Tract. in Ioa., 6, n. 9.</sup>

Baptism is invalidated by saying "In the names…" (plural).

Pohle-Preuss, The Sacraments: A Dogmatic Treatise (vol. 1): The Sacraments in General, Baptism, Confirmation, § "Matter and Form" of baptism:

Baptism would still be invalid if the minister would introduce a phrase embodying an anti-Trinitarian heresy,⁵⁴ e.g., “I baptize thee in the names of the Father, and of the Son, and of the Holy Ghost.”⁵⁵

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<sup>54. Tritheism, Arianism, etc.
55. “Baptizo te in nominibus Patris et Filii et Spiritus Sancti.”</sup>

Father, Son, and Holy Ghost are the names of the Divine Persons, not of the Trinity.

God is one

not in the oneness of a single Person, but in the Trinity of one substance. […] distinction in persons, unity in essence
^{non in uníus singularitáte persónæ, sed in uníus Trinitáte substántiæ. […] in persónis propríetas, et in esséntia únitas}
—Preface of the Holy Trinity

Does it imply that there is but one name for all the Three Persons, and that Father, Son and Holy Spirit are not individual names?

Sort of; it's not that the individual persons don't have names, but that God has one Name, and the formula is as it is to emphasize that there is one God.

However...

What is the reason for starting with 'in the Name' and not 'in the Names'?

...the best answer is because that's what Jesus told us to do.

"Name" here ("ὄνομα") is singular.