I agree that storing recovery codes is a pain point, but they're fundamentally different from passwords in that you don't need to use them for each login. That allows you to put them in cold storage, whether that's an encrypted flash drive, a piece of paper, a box buried in your back yard, or whatever else you want. Doing the same thing for information you need on each login would be ridiculous, but for a once-in-a-blue-moon recovery situation, the lack of convenient access is fine.