I did the whole “reauthorization” thing a couple months ago when I got a new computer and couldn’t remember all the devices I’d previously authorized, and it was super easy to set it all up again. Just a matter of deauthorizing all of them, then going around and clicking a button to reauthorize the iPods/computers that I still actually use.
Next time, de-authorize Itunes before the re-install, then authorize it after the install. Not always possible with a crash, but it keeps you from running up against the once a year limit.
I’m not surprised. When we set up separate users for the kids, I had to authorize each separate user as if it were a separate machine.
Also it periodically forgets, at least on my laptop; when I try to copy stuff off my iPod onto the laptop (using the “copy purchased items” tool), it says some can’t be copied. This despite the computer having been authorized at least once in the past for each of the kids’ iTunes accounts.
But yes, the answer to the OP is the “wipe out everything” process. Annoying that they won’t let you do this more than once a year (in case something glitches).
I know of two people who were able to call Apple Support and get a second full set of re-authorizations. If you just tell them what and why, they will help you. They just don’t want to make it too easy so that people can share with lots of friends too easily.
But if you have 1. home computer 2. kids computer 3. apple TV 4. work computer, etc. They understand that it is easy to use up your 5 if something goes amiss.