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?
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?