I actually solved the problem myself, so I'll say it here: It actually wasn't my firmware at all; somehow my SD card got corrupted and all my game data couldn't be read by the Switch. That's why the issue persisted even after restoring my NAND backups. That included my game carts because they had updates that were corrupted. Redownloading the games and installing them with Tinfoil worked. Even my save files were still intact, just not the games. Everything's good now, but thanks for all your help!You started off with a bad stock firmware. ReiNX is working just fine with me on firmware 6.0.1 (not 6.1).
Are you keen in using another CFW in the meantime while Rei is fixing the fault in this CFW with 6.1? I suggest you try RajNX.
I'm not sure whether it was because of it or not, but I suspect the problem was my unsetting the archive bits in hekate.