It really is all about prestige and status. Even in fields that are ostensibly about something else like math, that something just matters because it's tied to status. If you prove P =/= NP and no one believes you, you have nothing. And although this counterfactual isn't really possible, if you had a false proof that everyone believed, you'd get all the fame.

I feel like I need to chime in on this. There is some truth here, but it is very hyperbolic (no pun intended) to put it like this.

First of all, probably the major proof of the 21st century so far, the proof of the Poincaré conjecture, came from Perelman, a relative outsider in mathematics. This was still acknowledged widely. There are also multple mathematical results by hobbyists that have been accepted, e.g. in

plane tiling or even

knot theory.

When it comes to false proofs, I don't think they stand. The thing is, multiple people will come along trying to understand the proof in order to modify it for their own problems, and eventually someone will find the problem. That said, conjectures tend to be a bigger problem. I know of multiple that are kiind of believed because they're made by big names and seem reasonable, but there's not actually a proof yet and still a whole theory has developed around assuming the conjecture is true. I've heard such e.g. in symplectic geometry, but don't know the details.

Here is a paper discussing correction culture in math, and it definitely seems like there are still problems. But I don't like the sweeping dismissal.