No, you won’t see the word covenant in Matthew 5. However, you will see it in the following references where it refers to the kingdom of God (the church) which was established when Christ was crucified. And I erred in saying Christ put forth the requirements of the New Covenant in Matthew 5. I should have said he put forth SOME of the requirements. What he did in that chapter was to contrast the works of the old law which were legalistic with the requirements of the spirit under the new law of Christ.
Jer_31:31 “Behold, the days are coming, says the LORD, when I will make a new covenant with the house of Israel and with the house of Judah
Mat_26:28 For this is My blood of the new covenant, which is shed for many for the remission of sins.
Mar_14:24 And He said to them, “This is My blood of the new covenant, which is shed for many.
Luk_22:20 Likewise He also took the cup after supper, saying, “This cup is the new covenant in My blood, which is shed for you.
1Co_11:25 In the same manner He also took the cup after supper, saying, “This cup is the new covenant in My blood. This do, as often as you drink it, in remembrance of Me.”
2Co_3:6 who also made us sufficient as ministers of the new covenant, not of the letter but of the Spirit; for the letter kills, but the Spirit gives life.
Heb_8:8 Because finding fault with them, He says: “BEHOLD, THE DAYS ARE COMING, SAYS THE LORD, WHEN I WILL MAKE A NEW COVENANT WITH THE HOUSE OF ISRAEL AND WITH THE HOUSE OF JUDAH
Heb_8:13 In that He says, “A NEW COVENANT,” He has made the first obsolete. Now what is becoming obsolete and growing old is ready to vanish away.
Heb_9:15 And for this reason He is the Mediator of the new covenant, by means of death, for the redemption of the transgressions under the first covenant, that those who are called may receive the promise of the eternal inheritance.
Heb_12:24 to Jesus the Mediator of the new covenant, and to the blood of sprinkling that speaks better things than that of Abel.
And, yes, Paul was a Torah keeping Jew until his conversion as recorded in Acts 9:16. With his conversion, he ceased being a practicing Jew. The rest of Paul’s life after this moment was dedicated to spreading the gospel. How that made him a practicing Jew, in your words, is a mystery.