To be fair, many of the proof attempts that mathematicians do also have major flaws. Most get caught before getting published.
But that's the actually important difference. Mathematicians have the toolset and processes to catch the flaws, random people using Claude don't.
Trust isn’t a binary, and I can trust things I don’t understand enough that I can use them. OP was talking about needing to understand, which is quite a bit above the level of being able to validate enough to use for a task.
Unfortunately, it is not, and many of its attempts at mathematical proofs have major flaws. You shouldn't trust its proofs unless you are already able to evaluate them--which I think is pretty much all the OP is saying.