I think it was Terrance Tao on the Lex Friedman podcast recently that said that there are very often little mistakes in big proofs, but they are almost always able to be patched around. Its like mathematicians' intuition is tracking some underlying reality and the actual formalization is flexible. Yes sometimes digging down into a small mistake leads to an unbridgeable gap and that route has to be abandoned, but uncannily often such issues have nearby solutions.
Also a lot of errors would be called "typos", not errors. Such as some edge cases missing in the theorem statement which technically makes the theorem false. As long as there's a similar theorem in the same spirit that can be proven, that's what the original was all along.
