So, I just took some time over lunch to try to reproduce your problem, and can’t. Here’s what I did:
Burned the latest version of Raspberry Pi OS ARM64 to a USB thumb drive.
Booted and stepped through the normal first-run experience (set timezone / keyboard / wireless network, installed updates, rebooted).
Installed Flatpak and Obsidian.
sudo apt install flatpak
flatpak remote-add --user --if-not-exists flathub https://flathub.org/repo/flathub.flatpakrepo
# Rebooted here, just to be sure
flatpak install --user flathub md.obsidian.Obsidian
flatpak run md.obsidian.Obsidian
# Note that the Flathub version is out-of-date, but will
# auto-update to the most recent version the first time it's run.
For good measure, tried to download and run the most recent AppImage.
curl -L -O https://github.com/obsidianmd/obsidian-releases/releases/download/v0.12.15/Obsidian-0.12.15-arm64.AppImage
chmod +x Obsidian-0.12.15-arm64.AppImage
In both cases Obsidian ran without issue for me (though FWIW the current 64-bit build of Raspberry Pi OS would hang on me every other boot).
You might try removing and re-adding the Flatpak repos (both user and system), but given that you’re also having trouble with the AppImage I suspect that won’t help. If I was in your shoes i might try burning a new Raspberry Pi thumb drive / SSD / microSD / whatever you’re using and then give it another go, as at this point I’d be suspecting that something was wrong with the underlying OS.
Sorry I can’t help more.