I don’t understand what problem they are meant to solve. If you have a FOSS piece of software, you can install it via the package manager. Or the store, which is just a frontend for the package manager. I see that they are distribution-independent, but the distro maintainers likely already know what’s compatible and what your system needs to install the software. You enjoy that benefit only through the package manager.
If your distro ships broken software because of dependency problems, you don’t need a tool like Flatpak, you need a new distro.
For simple applications this is probably not that wild. But the more complex programs we talk about, the more helpful are these formats. Programs like OBS or Firefox in example is a lot of trouble to compile quickly. And imagine more of these programs. Package maintainer of your distro could use the time in a better way. Those who want to package it themselves (probably Arch) could still do, but most who want to provide the newest Firefox could just use Flatpak, coming directly from the developer day 0.
One also does not need to wait until its packaged by your distro maintainer and it comes directly from the developer instead (maybe). The original developers often do not support all distros and would like to have a known state and version of the program that they can rely on, like a Flatpak.
That being said, I don’t use Flatpak. But I used it in the past and it was helpful in some cases. Even on an Arch based distribution. Currently I use an AppImage for a program that is not in the official Arch repos. The AUR has it, but the -bin is outdated and the -git version building from source takes too long and power. Even on my new modern machine it would take at least an hour for every new version. Or I just download the Applmage once (88 MB) and use the self updating system of it (which downloads newest version automatically and renames it to current executable filename). I’m talking about RPCS3 emulator.
All of this is right, plus AppImage provides portability. I have all of my emulators in AppImage format in portable mode on a portable drive so I can move it from one PC to another and have all my games, configs, saves, etc.