Unfortunately that breaks down when someone doesn't set multiple keys as backup and gets locked out. Then you're right back to password/backup code or some kind of recovery to email or phone. Chances are people just store their backup codes as plain text too. They also break down across desktop/mobile, e.g. register on desktop then try to log in on mobile. Not everyone has a good sync solution here, especially the non technical.

Honestly all the solutions have trade offs in UX/security/privacy and dependency on third party services. The best solution is going to be highly dependent on the business.