Here's a guide for how to set up SMS-free two-factor authentication on your Gmail account. It will cost you $18; if that's a hardship, contact me. https://techsolidarity.org/resources/security_key_gmail.htm
Unfortunately, the security key FAQ contains no info on why you shouldn't have your phonenumber. I assume it's because phone-number migrations can be used to take control of your accounts?
EDIT: Yeah. Perhaps reading the article instead of just coming to the comments might have been useful.