I think I also had this issue using Cinnamon once, but then I just used VLC instead. Never bothered to look into why. Worked fine in GNOME for me though.
I found a thread with a similar issue: https://forums.opensuse.org/t/flatpak-mpv-broken-since-20240306-snapshot/172981
There it seems the issue was audio. Try running the flatpak version with flatpak run io.mpv.Mpv --ao=pulse path-to-your-media-file
You shouldn’t have to install any flatpak dependency manually. Flatpak should handle it for you automatically when you install your programs. (In most cases.)