I was not familiar with the mathathon, and I suppose I am ignorant of the circumstances in which Tao allegedly bullied students into changing the event. I don't personally have any reason to doubt the veracity of the "Joint Statement about Mathathon", but perhaps you have some insight that I don't. I will say that I would imagine that any input from Tao, even in passing, probably has outsized influence on the organizational direction of the event due to his preeminence, but I'm not so sure that necessarily rises to the level of bullying.
But from what I gather, the theme of the statement is to expand the scope of participation to include work that is quite literally the /opposite/ of gatekeeping, i.e. taking proofs that are popularly understood to be opaque or difficult to understand (from any source, AI, human, or otherwise) and developing a new way to communicate them that is more accessible to more people.
I do see how it may come off as a slight to the original mathematicians who developed those proofs. But developing new ways to communicate already established concepts should not diminish the original works. There are probably much better communicators/teachers of relativity than Einstein ever was. No one out there thinks any less of Einstein because of Sean Carrol's "Spacetime and Geometry" textbook.
And this to me reads as only embracing the use of AI in mathematics, which is not gatekeeping behavior. Tao continues to emphasize that once proofs become effectively automatable, deeply comprehending and being able to transmit that understanding to others will become an increasingly important aspect of the field, the natural way for mathematicians to use the time that was once occupied by developing the proofs themselves.
Literally all of this is about expanding the accessibility of mathematics, which once again, I perceive to be quite the opposite of gatekeeping. Maybe we have different definitions of that term.
Also I have not seen Amadeus, so no comment about Salieri.