Now when I sign into them, after I enter my 2FA code it'll ask me for a 3rd factor - it'll ask me "Verify it's you. Google wants to make sure it's really you trying to access" and then I have to do something with an email address or phone number.
How can I permanently disable that?
I want 2FA, not unexpected 3FA...
Everything Google does has privacy invasion baked into it. Lots of people will readily enter their phone number (which positively identifies them personally) or they will enter an email address that Google hasn't collected before --- either of which will further Google's privacy objectives --- not yours.
Just say "no" to Google games.