There’s actually no evidence of a written Mishna, and plenty of evidence for the lack thereof. Last Daf Yomi round, I jotted down some notes on this as I made my rounds.
Before Rebbe it was like I wrote above about was it was before Rav Ashi. They always studied Talmud — the reasoning and proofs of the Halachah as well as Halachic statements. Every Rebbe taught Mishnayos to his Talmidim. This is the summation of the Halachos that he received from his Rebbes or had ruled on. This led to a situation in which any two people had a completely unrelated Mesora.
Rebbe brought everyone together. This was a tremendous undertaking. As it says, only Rebbe was able to pull this off with his wealth, government connections and peaceful time period. He had everyone recite what they had received from their Rebbe. Then, he painstakingly worked everything out and formed one solid, coherent Mishnah that should be the standard. From then on, that was what every child was taught and memorized.
Two more points. It appears that the titles of Mesachtos predated Rebbe. We find in the end of Horios how Rabban Gamliel organized Uktzin overnight. This tells you that it was a topic with a title (unless the Gemara was speaking shorthand), and that there was no set Mishna yet.
The other point is that the Gemara says in several places that the Mishna is Rebbe Meir according to Rebbe Akiva. This means that Rebbe used Rebbe Meir’s Mishna as his basis. On top of that he could have rejected parts and added even large sections.
Even if you’d say that he went ahead and wrote it, that was not his monumental achievement. But he didn’t.