As you note, we already have a system that uses more appropriate cryptography (than a PAKE) to solve this: FIDO.

You've lost me at mTLS here. At some point it starts to feel like we're advocating for security protocols just so we can fit them all in somewhere.

That was a bit tongue-in-cheek, sorry. I've worked in mTLS shops and it's definitely not practical for the public Internet.

Ultimately, I think the practical solution to homoglyphs is in the presentation layer, whether it be displaying different scripts in different ways, warning when scripts are mixed, or some other kind of UX rather than protocol change. The only protocol change I can think of to address them would be to go back to ASCII only (and even that is more of a presentation issue since IDNs are just Punycode).