You only get locked out of a website if you delete your cookies while your email provider is down. How often does that happen?
This idea doesn't speed up the login process, but it accomplishes a few other useful things. The server doesn't store passwords, so a breach of the server doesn't compromise other services for which users had duplicate passwords. And users can't compromise their own accounts by choosing weak passwords. Both scenarios are commonplace.
To be fair, the user can still compromise all their accounts by choosing a weak password for their email account. It does reduce the onus on them from coming up with dozens of (hopefully) unique, strong passwords to one, which is certainly an improvement.
This idea doesn't speed up the login process, but it accomplishes a few other useful things. The server doesn't store passwords, so a breach of the server doesn't compromise other services for which users had duplicate passwords. And users can't compromise their own accounts by choosing weak passwords. Both scenarios are commonplace.