The distro I use (Pop_os) has a lot of outdated packages in the 'store', & I think that's pretty common for any distro that's not bleeding edge.
The store compensates with flatpaks a lot of the times, but my issue with those is they don't integrate with the system very well. To launch Krita (for a random example), instead of using 'krita' in the command line, I need to use like, '~/.local/var/org.kritafoundation.krita.flatpak' (I am making that up, but it's probably something similar). This becomes a real issue when using scripts or needing software to work system wide.
Appimages are actually my current favorite method for universal install. I rename them, then stuff them in my ~/bin/ directory. My gripe with appimages is there is no auto generated .desktop file.
I like building from source, but sometimes it feels like I install 10 gigs of dependencies to build a 400kb piece of software.
Adding repos was invented by the Devil & is thus appropriately cursed to break systems, so I stay away unless there's really no other options.
I guess what I'm saying is every single method sucks, but still better than Windows lol.
Honestly, an up-to-date store would be the best of all Worlds, which I suppose is Arch's AUR, but, nah. Arch looks like a headache.
Anyways, I'm blabbing. That happens when I start geeking out about Linux.
I'm using Cosmic, which apes Gnome but is not Gnome. I'm being a total purist though & keeping as much KDE & Gnome out of the DE as possible.
That said, Cosmic has no problem updating the menu when Flatpaks are installed. I also don't tend to run software from a script just for funsies, it's usually to do with an automated process. Another issue is sometimes, a script will call on a hook for software, like a browser for example, & flatpaks can cause problems with the scripts locating or sharing data with the scripts due to the nature of the sandboxing.
I still have plenty of flatpaks installed because they are pretty good, even with (what I consider) flaws. Imo flatpaks are kind of like Docker in that they're great & everything, but also kind of a pain in the ass.
As far as appimages, I guess it doesn't bother me to update manually once in awhile and, as I just explained, I actually usually want software to have access to everything. I installed Ghostty via appimage, for example, & I use it all the time, but it's not critical that it gets updates, so I just check manually on occasion.
Anyways, I 100% agree with you on the rest. I've never considered the AUR as a giant repo but yeah, that's a good way to think of it.