I prefer the AUR, but if I have to use one of the three it's gotta be an AppImage these days.
I used to swear by flatpak, but because I'm on nvidia it just turns into a stupidly bloated mess since it never removes older driver versions.
They're certainly not "bad" though, and I use them on my SteamDeck for sure.
I preffered AUR over flatpak before as well but that changed more and more every time a AUR package didn't launch anymore or didn't update anymore because of some build error. I actually started migrating a lot of stuff over to the flatpak version because it just works.
I've had pretty much the opposite experience most of the time. Constantly dealing with flatpaks not opening or other weird issues that would go away as soon as I tried the AUR version or just installed it manually from the repo.
Weird. I've never had an issue with flatpaks, which makes sense since it's exactly the same on all systems. If it didn't launch for you, it shouldn't have for anyone else either, so the issue should have been fixed before the developer released a new version.