It's hard to avoid ever opening iTunes Radio and who wants to nuke their phone and start from scratch? After hours on this, here's my recommendation:
0. Open iTunes Radio and play a station. This loads Artist profile images from iTunes, and forever afterward music app will use iTunes artist pictures for exact name matches.
1. Create a playlist with 1 song from each album you want to use as the artist image.
2. Erase all music from iPhone. This clears non-Radio artist image cache.
3. Sync only this playlist.
4. Open music app, see which images you like and which you don't.
5. Change "albumartist" tag for this artist. I add a "." to the end instead of a space so I know which artists I've done this for.
6. Make iTunes reload the files if you used another program to change tags (I use mp3tag). Add Folder might work, idk.
7. Erase all music from phone.
8. Sync artwork playlist.
9. Open music app and scroll through all artists. This is very important because music app creates the image for all time the first time it is displayed (not when first synced!). If any album comes up with blank tile, quit music app using double tap on home button and restart it. Music app wants to keep scrolling smooth so doesn't always generate a tile and it won't until restarted.
10. Sync the rest of your music.
This works great if you have Group By Album Artist turned on. I'm not sure why you would ever turn this off since then your Various Artists and collections are all over and there's a bunch of artist entries with 1 song, but whatever if you don't enable this maybe you have to add a "." to artist tag then?
Most of the iTunes pictures are terrible compared to your choice of the best album art, but some are good and this way lets you choose without having to fear accidentally touching Radio or some update from Apple. If you want to change art later on you can unsync/sync one artist at a time like above. It's just not worth it to try to keep the phone in some kind of pristine state.
It blows that you can't just select Artist artwork in iTunes and leave all the tags correct, but at least this works.