So instead of letting the system, which can easily determine which auth system to use based on the email, you're putting the responsibility on the user to remember what they used when they have potentially tens/hundreds of web accounts? Yet somehow putting the password on the next page is worse??
If they're going to use a password manager then the password manager will tell them. Your system can't determine if the user signed up with Google or is using an electronic id card either.
If you're using an electronic id card then you don't have the same username and password values that your website uses. You likely won't even use the same fields. Authenticating through that usually just involves a separate button.
Auth systems don't key off of usernames... When we added FIDO support there's still a backing user store that records user x has FIDO y, with a recording of the public key. That's how the pairing works. You can see it today on login.live.com
