I think we should move away from passwords entirely, the UX for that is often much better for users (and as a bonus, you don't need to worry about your password being compromised, or password managers).
The US Government got rid of passwords in 2004 by switching to smartcards, and Google also did so in 2017 by switching to hardware security tokens.
2. The core problem is still that you need to be able to communicate with others, for which you need to identify yourself. Where a site is served from doesn't change that.