Haleivi,
There is plenty of evidence that Rebbi wrote down the Mishnah. Primarily, the mesorah of the Rishonim (e.g. the Rambam) that Rebbi wrote it. And although Rav Sadya Goan is ambiguous, the better reading (IMHO) is that he supports the Rambam’s mesorah.
But my point was that you see that there was a proto-Mishnah that was organized to a substantial degree before Rebbi. Not just by topic but in specific wording. There are machlokisim within a mishnah about what the stam portion of the mishnah means. That means the Tanayim–in generations before Rebbi–were operating off the same text.
So it could be that Rebbi simply organized/composed a universal wording incorporating the disputes of the recent generations. But then his work wasn’t all that innovative. He was just continuing a process that Rabban Gamliel (and others) had done in previous generations. But if you say that he wrote it down, changing Torah Shbal Peh to being written, it was truly monumental and worthy of reputation Rebbi has in our mesorah.