Maybe there's a sort of golden middle way here where all mathematical prose-proofs should be annotated[1] by the associated computer-checked proof. The trustworthiness of a particular proof could be assessed by the number of such references and how much coverage (of the prose) they provide...?
[1] Perhaps only by reference, as here. :)
EDIT: Quick edit, I say this as someone who -- earlier in xir career -- probably subjected a lot of people to somewhat verbose comments. In practice, my comments were usually right and the programming language wasn't powerful enough to capture the semantics of what I was doing. One hopes this is the distinction between good and bad comments. Sorry for veering off-topic.
It does make me wonder, though, about those computer-generated proofs which are so massive that no human can understand it. If you can run it...?
A proof is a proof if you’re convinced. Ideally, a proof is correct, but that’s not always the case.
(Except for undergraduates! Show your work! ;))