Computing, on the other hand, is plagued with broken programs, and formal methods are still a very specialized area, even though the potential benefits are much greater in computing than mathematics.
But you don't know that.
There certainly were precedents: https://mathoverflow.net/questions/291158/proofs-shown-to-be...
Obviously the people who like proof assistants disagree, and they may be right. It could be that modern theorems are just too complicated to be adequately evaluated by unassisted humans. But mathematicians have been getting along just fine without them for a long time, so "why aren't we requiring computer-checked proofs of everything" has a very straightforward answer: mathematicians writ large aren't convinced of the benefit yet.
I guess the question is which of "convincing a human" or "convincing a computer" is considered the higher bar.
In any case, uncovering "what you believe that isn't so" is pretty important.
There's obviously been a lot of work done on proof assistants and maybe it will become easy and natural to formulate most proofs in a way a computer can check, but it's not anywhere close to that yet. The book explicating the proof discussed in the article is already 500 pages long, translating that into something a computer could check just isn't at all feasible yet.
Because mathematics has more or less worked fine for decades without it, and most mathematicians aren't programmers and aren't interested in it. Proofs are for understanding, not for technical correctness (though obviously the latter is very important). A good proof convinces someone that it's true; convincing a computer that it's true is probably tangentially useful, but not very interesting to most mathematicians. They mostly want to be expanding the frontiers of mathematical knowledge, not painstakingly encoding proofs that they already believe (and, crucially, that their peers also already believe) into a computer system.
That's not to say proof checkers aren't interesting in their own right, but there's lots of really quite good reasons why mathematics goes without them almost all of the time. Formal methods bring a lot to computing, where our programs fall apart constantly, but mathematical proofs almost always don't fall apart. And in computing, almost all programs don't have formal proofs of correctness! If any area needs them, it's in computing, not mathematics. You might as well ask why every compiler doesn't come with a proof of correctness: because it's already hard enough to build a compiler, let alone the monstrosity that would be a formal proof.
The added benefit to a computer-checked proof is usually small to nil, and it's a ton of work. Like, way more work than just writing the proof.