It would appear that you have to disable the old Two-Factor authentication to allow users to continue to log into using the legacy shotgrid method.
At the login page the “Sign in with Autodesk ID” button works as expected, with or without two-factor enabled on your Autodesk ID.
If you use the “Sign in with ShotGrid” button to use your legacy login, it takes you to the normal username and password fields. When you log in, it always takes you back to the two buttons letting you choose which login method to use. I think it is checking the credentials as sometimes I would get a warning telling me they were invalid but other times it would not show anything. I’m guessing it accepted my credentials but didn’t handle the redirect to the two factor authentication and failed the login.
Once I disabled two-factor authentication in the site preferences I was able to log in using the legacy mode. The option to force two-factor on the Autodesk ID did not seem to make a difference.