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.
I think he accounts for that in the answer, "in a way that would be faster than if one or both proofs were informal" (which assumes "one formal and one informal" is also a case he's talking about). The way I understand his point is that in either case you would have to go through the two mathematical structures with the same amount of rigour and attention until you find the divergence, and that's easier to do when at least one side is formal (but can be done in either case).
In other words, informal math doesn't make this problem easier because you can still make and miss mistakes in encoding intent into the structure. But at least with formal math, there's whole classes of mistakes that you can't make.