Based on what we know of the history of the very early church, it wouldn't have been easy for them to coordinate around making changes to the story - a centralized authority didn't exist until hundreds of years later. The total fabrication theory doesn't have a clear starting point for when everyone got together and decided to make up a character. That in my mind lends some credence to the idea that there was an individual that they could gather around.

If you look at how cults or religions got started in recent history, we never see a bunch of people coordinate in that kind of a way, where a lot of people disparately decide to invent a nonexistent leader and go around telling everyone that they recently died.

Could the central authority have later fabricated a history of the early church that made it look like there was a long period where nobody could have coordinated big changes? Maybe, but that is a standard that no set of historical documents could pass. (It's like suggesting that the Byzantines fabricated every single classical document including the tree-rings of ancient historians arguing with each other.)