I don't think I've met anyone who learned to write a good proof without significant human instruction. That is probably something to be expected because much of what makes a "good" proof is social expectations, but I see a lot more cases of unjustifiable leaps of logic than excessive, obvious detail.
I feel like it’s relatively easy to learn a lot _about_ math, as a self learner, without knowing a lot about how to _do_ math.
I’m mostly set taught, and I know the notation, I know the vocabulary, and I can follow along with a lot of math papers without too much of looking up terminology and I’ve adapted things I’ve read in math papers into working code, but I’m also well aware of my limitations —- I don’t know how to evaluate whether any of these papers are valid, I don’t have any sort of working knowledge of the tools of proofs to be able to do anything outside of fairly elementary calculus, etc. I know about higher math, but I wouldn’t say I know higher math. But even without that working knowledge, I wouldn’t say that what I know is valueless.
I hate the "writing math on windows thing".
I agree its somewhat disingenuous, and not representative of how mathematics is really done (most of which regular people would probably find pretty un-exciting and mundane). But it does increase visibility about those things. Instead of John Nash being someone who only mathematicians/academics knew of, his name is now (somewhat) more better known, isn't it? A book and a movie.
To be honest, I grew up in a pre-internet, pre-wikipedia world, and without movies like those I maybe would have never heard/known about these folks and what they did.
Hollywood isn’t making movies about “tortured geniuses” to pander: they’re doing it to reflect reality.