I personally view proof checkers as mathematicians' tools so I assume mathematicians would be involved either way. With some percentage actually preferring to work closely with these tools. See also Terence Tao's comment in https://terrytao.wordpress.com/2025/05/31/a-lean-companion-t... which feels relevant to me
Tao misunderstands the question here: it was about reconciling a traditional (informal) proof and a formal proof which come to opposite conclusions, not about two different formal proofs.