Isn’t the problem when the login dialog pops up? I think the password managers have a hard time with that login dialog (which the browser itself throws up for certain AUTH scenarios, like if you were using .htpass or something old school). My Orbi router does the same thing and I have the same problem with 1Password there. If there were an actual login page it would be fine. caveat: I have not looked at the source code to see what part of the stack implements the auth dialog.
I too would like the Home Assistant Login screen to work with LastPass. Suggestion: bundling together these two features in one request seems unwise. Updating the login screen to be more friendly with LastPass is likely trivial, just need to make sure the field are easily identified with unique names. Integrating with Single-Signon is likely more difficult. Worried that bundling these two together means the easier fix won’t get done