Getting a proof, even in one that is amenable to computer verification into a form where you can actually verify it is a monumental task, like it can easily take multiple experts multiple years.

Adding to this problem the intersection of mathematicians who understand computer aided verification and who understand a particular piece of research mathematics is often empty, and even if there are a handful of people who could do the necessary work they're often busy doing more interesting stuff.

Finally I don't think a big proof that has been accepted by the community has ever been found to be false by programmatic methods. I think its unlikely that this will happen for a long time, for the simple reason that to feed a proof into a computer requires someone to dissect the proof to its logical bare bones in the first place, and the person doing the dissection is likely to notice any errors before they get round to feeding the problem into the computer.