Syncing of "MFA codes" is really syncing of the secret component of TOTP (time based one time password).
And it's a good thing, and damn any 2fa solution that blocks it. I don't want to go through onerous, incompetent, poorly designed account recovery procedures if a toddler smashes my phone. So I use authy personally, while a friend backs his up locally.
Hell, no bank I use (several large and several regional) support generic totp. Some have sms, one has Symantec VIP, proprietary and not redundant.
Edit: since I'm posting too fast according to HN, even though I haven't posted in an hour, I'll say it here. Symantec is totp but You cannot back up your secrets and you cannot have backup codes.
A better way to fix this is to have multiple ways to log in. Printed backup codes in your safe with your personal papers and/or a Yubikey on your keychain. This works for Google and Github, at least.
Passkey syncing is more convenient, though, and probably an improvement on what most people do.
If you can backup a key it is not MFA. It just a second password and not another factor. The solution to having your phone smashed is to have multiple "something you have", so you have a backup.
And it's a good thing, and damn any 2fa solution that blocks it. I don't want to go through onerous, incompetent, poorly designed account recovery procedures if a toddler smashes my phone. So I use authy personally, while a friend backs his up locally.