It is indeed the very epitome of sanity, if you simply consider that the codes are secrets, and this entire practice is derived from having hardware dongles with secure enclaves, where secrets go in but never come out. It is the utmost in security when this one-way relationship is observed.

The ability to export secrets is an unfortunate compromise which vendors make for consumer markets. The MFA apps were not designed for exportability. If you own any Yubikeys you will know this. The whole idea is that this factor is "something you have", in other words, possession of the item containing your secret. An exported secret is no longer a secret, no longer something you have; it's just another password you're shuffling around.

The reason that you don't lose access to accounts when losing your MFA apps is that you took down the emergency backup codes and you committed them to paper, or some other durable medium, in a place where they can easily be accessed during a crisis. You did this scrupulously with each MFA activation, didn't you? Didn't you?

Typically the way these codes are compromised is when they are stored in a non-HSM location like Google drive or transferred somehow. Then again, if you are just trying to keep people out of your Facebook account it's not a big deal. But if you are trying to keep people from financial accounts I wouldn't recommend transferring TOTP keys. Instead using a backup method like a printed out one time use sheet would be better.

Unfortunately most such websites use KBA or Text based authentication as a backup for TOTP so you may as well just stick it in Google drive.

Yeah, the iron triangle of security, convenience, and privacy rears it's ugly head again.

I think Microsoft Authenticator is the smartest right now because it's a "two-cloud" solution partly out of necessity, but also that seems a trustworthy architecture more generally. Since almost no one's phone runs Windows anymore, the raw app data backups "naturally" go to either iCloud or Google Drive. Then Microsoft keeps other (HSM) decryption keys in OneDrive. The threat model requires compromises of two clouds, so Microsoft Authenticator can be way more generous on how often and easily it backs up. It's an interesting point in the security vs. convenience tradeoff.