If I am correct I do remember that ' back in the days.' you can still authorize iphone devices and Macs and PCs. There is no solution that answers this question specifically.
You can! You can authorize Macbook Pro to Imac, Imac to Macbook Pro, Mac to PC and vice versa. (Although they are "alternative solutions" around "authorizing" content on other devices other than Macs) it seems that Apple does have a drawback on this matter. You know after all the trouble I went through myself it seems that the solutions shown above. We have our phones in our hands, reps aren't robots, their people. You should also try being polite to people, you seem like you're not correct too often with having to point out some reps inability to blindly resolve an issue with someone's phone over the internet. home sharing has my APPLE ID and I am still not authorized. That way there what has been tried can be told to the rep. It's also great if before someone pressed that option if they'd take note on a pad of paper. Working over the phone or internet is tough, it's our descriptions that allows them to assist us. I am sure if a rep had our phone in their hand they could have figured it out much quicker than you did.
So as home sharing worked for you that's great. Each individual's issue could be different depending on what they attempted while trying to authorize their phones. It's more like kind of strange a rep doesn't know all the options or settings a person has entered or changed in their phone.