You have outdated information. There are no longer any tradeoffs to AppImages:
Yes there is no “official” default installation path, but like how XDG_DATA_PATH isn’t technically a standard but practically it is, the de-facto standard is ~/Applications now, and most AppImage-based tools respect that.
They integrate fine with the system. Better than Flatpack and Snap, actually. I’ve had lots of issues with flatpaks not respecting themes, but never AppImages. Not sure where you got that from.
I solved the other problem with AppImages with a package manager I wrote. Centralized location pointing to AppImage urls, and it downloads and keeps them updated. And no, you don’t need to write your own, there are multiple AppImage package managers out there.
On the flip side, there’s no weird extra locations like how flatpak installs apps, you know exactly where the program is in case you want to launch it manually, you can mix apps available in your package manager with ones you download directly seamlessly, no dependency hell or version problems as AppImages are self contained (even multiple versions at the same time), etc, etc, etc all the benefits people spout about AppImages.
AppImages imo are the superior cross-platform package format as there are no tradeoffs and no downsides, meanwhile:
Snaps are slow and proprietary
Flatpaks suck to create and maintainers select all on sanboxing, so it’s a complicated mess for no reason, and they also have bad theming that never works half the time.