A well-designed MFA system has multiple fall backs available. Google's has options to fall back to sending an SMS (not the highest level of security, but security is always a trade-off), so if you're phone's broken you can move the SIM to another one. Or a phone call to a pre-registered landline that will read out a code to you. As well as the paper/printable emergency codes.

That said, there's always going to be compromises and annoyances - I've been using MFA for 10+ years, and occasionally something glitches out, but other things (e.g. basic password systems) also sometimes glitch out, and having an account compromised is much, much worse than a minute or two of mild annoyance.