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Ephesians is a letter for the Christian church (a variety of home churches of people from a variety of backgrounds). Ephesus had a large amount of variety in religious backgrounds and beliefs. From Introduction and Background: Paul's Letter to the Ephesians,
"Ephesus was pluralistic in every way. Many ethnic and cultural backgrounds were represented, and religious pluralism was entrenched and embraced."
This letter's aim is not converting Jews to Christianity, but guiding new Christians of eclectic backgrounds.
"[H]e does write into a culturally chaotic, temptation-laden environment in which his implied purpose is to instruct and encourage Ephesian believers in their calling to unity as the Body of Christ, both Jew and Gentile, and in their growing in Christian maturity." (ibid.)
The emphasis on 'one' at Ephesians 4:5 is similar to Paul's statement at Galatians 3:27-28.
"For all of you who were baptized into Christ have clothed yourselves with Christ. 28 There is neither Jew nor Greek, slave nor free, male nor female, for you are all one in Christ Jesus."
For the Christians in Ephesus, they need to recognize they all have one Lord, one faith, and one baptism. They are all one in Christ. This says nothing directly about whether the Mosaic Covenant is still binding on certain people (i.e., Jews who have or have not converted to Christianity).
Compare Hebrews 8:13.
"By speaking of a new covenant, He has made the first one obsolete; and what is obsolete and aging will soon disappear."
'Will soon disappear' means it still exists. When will it disappear? AD 70, with the destruction of the Second Temple, is an obvious possible point.
Compare Matthew 5:18.
"For I tell you truly, until heaven and earth pass away, not a single jot, not a stroke of a pen, will disappear from the Law until everything is accomplished."
Jesus is saying the Law will be upheld until 'heaven and earth pass away'. When does that happen? For many preterists, an obvious date is AD 70, with the destruction of the Second Temple.
Similarly, Matthew 24:34-35.
"Truly I tell you, this generation will not pass away until all these things have happened. 35 Heaven and earth will pass away, but My words will never pass away."
There is a straightforward interpretation of these lines. Heaven and earth will pass away within 'this generation'. That is within about 40 years after Jesus spoke the words. AD 70 is in that range (general consensus is Jesus was crucified in AD 33).
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The "Old Covenant" was a deal between God and the physical nation of Israel. Israel agreed to follow God's laws, and in return they would receive physical blessings. But if they didn't they would receive physical curses. The Israelites were "chosen" for the purpose of providing a physical example to the world of the effects of obeying God's laws.
The "New Covenant" was a deal between God and the spiritual nation of Israel. Christians agreed to follow God's laws, now interpreted with a new spiritual meaning, and in return, they would receive spiritual blessings. True Christians were "chosen" for the purpose of providing a spiritual example to the world of the effects of living God's way.
Jews today (and anyone that converts to Judaism, regardless of ancestry) can still become part of God's physical covenant ("Bar mitzvah" means "son of the law").
And anyone (including Jews) can become part of God's spiritual covenant.
There is nothing incompatible between these two covenants.