The scenario you're describing would be the full Section 4 process, when the president disagrees that he's incapacitated.
While theoretically possible under that amendment, I don't believe Biden would have fought it out. He simply couldn't. He was and is simply unable to make a coherent case for remaining in office.
I also believe that his handlers and staff would have bailed (or be "asked" to bail), leaving him to be just a stupid, senile man on his own. Without speech writers, people to administer his drugs, or maybe even people to change his diapers, what could he have done? Walk over to the Washington ComPost and clearly articulate how and why he was capable of running the executive branch by himself? Impossible.
All that was certainly presented to him, which is why Pelosi got involved in the first place.
So without the option of fighting it out under Section 4, then it becomes a straight-up replacement under Section 1. That means he, Jill, and Hunter would have been thrown out onto the sidewalk and the diversity hyena would become president. That was Joe Biden's 25th amendment, and he couldn't let that happen.
So pulling the plug on his reelection campaign was what he had to do.