I think the demand for "tangible" proof (if by that you mean, fully mechanically-verifiable proofs and a style unlike anything common in mathematics papers today) is a bit silly and seems to be driven by some ideologies well outside of mathematics, rather than mathematicians themselves.

A proof is whatever convinces sufficient mathematicians that the theorem is consequent! Classical logical systems are a very good way to do that so they get used a lot. But they're not the only way, and involving a computer program makes most proofs less convincing rather than more.