Are there any stable AOSP ROMs for this which look stock? Is CM10.1 stable enough?
I would say so yes, I was using the last CM10.1 unofficial nightly, they've gone over to official ones now which ironically I think were less stable to begin with but that was a few days ago so I expect they've sorted that out now.
I couldn't find anything NOT working on my S4 with the last CM10.1 I used which is quite a revelation in comparison to previous CM builds.
One thing to note is that some kernels, and I never really spent too long figuring out which, knock out the wifi on a reboot but it's a very simple string you need to enter into the terminal emulator on the handset to get it switched back on again and then it stays on until a reboot.
The final very niggly issue I had was that my carrier came up as something like 23654 not vodafone, thats also easily changed so it ended up saying Matts Galaxy S4 instead of Vodafone
Very small issues and if you're on the official nightlies you can just keep flashing the latest over your old version.
So I tested.. Wifi, 3G, bluetooth (including bluetooth calling which is often an issue), camera, gallery etc etc.. the only thing I didn't test was NFC (but who cares right?)..
After all that, I came back to wanamlite with jb domination because once you've turned the greens to blues and gotten the stock font, some of the samsung extras are actually worth it (smartstay for example) plus whilst the CM10.1 battery life was fine.. I could get to the end of a day just about (I use my phone for work so a lot of screen time and calls) I can breeze past a day with wanamlite, it's very impressive.
I came to the conclusion that on balance using wanamlite with nova launcher with the JB icon pack, holo locker (stock lock screen), stock messaging apk and jb domination was far enough away from the cartoony samsung software to make the extras worth it.