The Bavli doesn’t mention Chinuch as A reason. The Bavli, obviously had the Megilas Taanis, too. It is constantly referenced. The Bach quotes it and so did the Ohr Zarua. The print that I downloaded is a very old one. There is no reason to think the Beis Yosef didn’t have it.
There is no mention in the Megilas Taanis of the Yom Tov being Zecher Lachinuch. The Ohr Zarua, who mentioned the MT in passing, suggests that the name Chanuka comes from Chinuch, but doesn’t call it the reason for Chanuka. He quotes the Gemara as the reason for Chanuka. Then he quotes the other part of the Megilas Taanis to show that they were busy for eight days.
There is no reason to celebrate the process of cleaning up, which is what the MT describes as the function of those eight days. Someone became excited upon bumping into the Megilas Taanis, which has been around for years, perhaps always, and decided that nobody had it and he now knows the REAL reason for Chanuka.
If you have to tell me that the Gemara only had the Raisha — which it quotes word for word — in order to protect a theory, it isn’t worth protecting.
The Tur had the Ohr Zarua, at least. And yet, he still says that the name Chanuka comes from Chanu Chaf Heh. That’s how insignificant the Chinuch aspect was to the Tur as to the essence of Chanuka.