> I'm surprised Google encourages syncing the codes to the cloud... kind of defeats the purpose
Probably so when you upgrade/lose your phone you don't otherwise lose your MFA tokens. Yes, you're meant to note down some recovery MFA codes when you first set it up, but how many "normal people" do that?
A number of sites I've signed up for recently have required TOTP to be setup, but did not provide back up codes at the same time. There's a lot of iffy implementations out there.
With Google Authenticator some years ago it wasn't even possible to restore your codes even if you had a local backup of the device. I'm not sure if that still is the case today but it was a common issue which we saw at our service desk before we switched to a different solution.
Probably so when you upgrade/lose your phone you don't otherwise lose your MFA tokens. Yes, you're meant to note down some recovery MFA codes when you first set it up, but how many "normal people" do that?