That's your problem, your trying to "read into" it when there isn't anything in the verse that demands the Jews and Grecians be treated "collectively." The first sentence speaks of preaching to Jews only, and the second sentence speaks of preaching to Greeks. Period.
The tradition may be wrong, but that's not the point. The point is you can't prove it from the Bible.
“That’s your problem, your trying to “read into” it when there isn’t anything in the verse that demands the Jews and Grecians be treated “collectively.”
(Acts 11:19-21) “Now they which were scattered abroad upon the persecution that arose about Stephen travelled as far as Phenice, and Cyprus, and Antioch, preaching the word to none but unto the Jews only.
20. And some of them were men of Cyprus and Cyrene, which, when they were come to Antioch, spake unto the Grecians, preaching the Lord Jesus.
21. And the hand of the Lord was with them: and a great number believed, and turned unto the Lord.”
Notice the verse 21 says that the Lord was with “them”, not “some of them” as in verse 20. “Them” refers to the “they” of verse 19 and includes the “some of them” in verse 20.
As far as Peter is concerned, there is nothing in the text that would lead anyone to believe he had anything to do with the Antioch church.