But for OP, the point isn't having humans understand. It's having the computer check the proof. You still use traditional journal papers for communicating the proof to humans. You're just also able to say "The proof has been verified and really is true."

That's a good point... I agree the tools are not ready. Most computer-assisted tools make it easier to do your work. It's easier to type a five page paper in Word than it is to write it longhand. But for now it's a lot more work to encode a proof in a proof assistant than to just write it up in LaTeX (which I think a lot of people don't realize).

It's not just that the foundational concepts are missing, it's also that proof assistants currently require you to fill in much more detail than human proofs do. You can't just write something like... "this corollary is trivial", or, "since x is a power of 10, obviously cos(x^2) is irrational"; you have to spell out a proof.