Way for iTunes to respect my custom Artist/Album assignments?

THE EXPECTED PATH FOR A PROBLEMATIC SONG IS iTunes/iTunes Media/<ARTIST>, BUT THE FILE IS ACTUALLY IN iTunes/iTunes Media/Music/<ARTIST>...

Since the song file is located at the correct location, I would first try locating the song file for iTunes when prompted (at the correct location). If all of the other un-locatable song files are "problematic" for the same reason, I think iTunes will be able to extrapolate their correct locations based on the info you provided by locating one of them. I would try it that way. If iTunes cannot automatically locate other missing song files, try the method you described.

Good job in finding a solution that works, although it may be time-consuming to go through 1600+ songs (I hope most of them are on complete albums instead of individual songs).

It is true that Album Artist (when present) is used instead of Artist (for song file path). The Album Artist field was added because the Artist field can sometimes have an entry that is different from the primary artist of the album. For example, the album's main artist is Bob, but for one of the songs on album, there's a guest artist, and Artist field says Bob and John or maybe Bob feat. John. If only the Artist field is used for iTunes Media folder organization, the album's song files get split up into different folders. But for this example, the Album Artist field always has Bob for all songs on the album. iTunes uses Album Artist for iTunes Media folder organization (because it's present), and all the song files on the album stay together in one Album folder.

For a compilation album like a "Greatest Hits" collection (with no primary artist), using Various as Album Artist works to keep all song files of the album together. The songs are organized using Various as the Artist folder (instead of the actual artist), but at least all of the song files are stored together in their respective Album folders.

If you don't plan to have a different entry in Artist for songs within an Album (as a general rule for your re-organized iTunes music library), you can clear the Album Artist field as you go through your library to edit the Artist entry. When you are done (with the "project"), you can sort your music library song list using the Album Artist column, to see if you missed any. Then iTunes always uses Artist to organize the iTunes Media folder. OR, you can use the same entry in both Artist and Album Artist fields (as your general rule), which allows you to have useful exceptions when you want to preserve differing Artist entries within an album (and also for compilation albums with no primary artist). Songs purchased from the iTunes Store have entries for Album Artist.

In my music library, I have songs with and without Album Artist entries. My (not very strict) rule is, if it's already there, I don't remove it. If it's not there, I don't add it, except when I notice that the Artist entries for an album (with more than one song) is not the same for all songs.