My opinion is Paul gave up on the Jews as a group because he devoted his ministry to the gentiles. Paul knew that Jesus didn't give up on the Jews as a group and that "one day all Israel would be saved." That's in Romans 11?
The most interesting way Paul proved that Jesus is the Messiah is he praised God the Father and the Lord Jesus Christ in the openings of all of his letters. Paul respected the godhead in the OT where there is YHWH God of hosts, YHWH of hosts and the Holy One of Israel (Jeremiah 50-51 and Isaiah 7-10).
Curiously, Paul didn't include the Holy Spirit in his openings.
And Paul in Colossians and the writer of Hebrews call Jesus the firstborn Son who was chosen (Luke 9:35).
This fits the archetypes of Shem, Abram, Isaac, Jacob, Joseph, Ephraim.