Is the gnome-keyring working?
https://wiki.archlinux.org/title/GNOME/Keyring
//EDIT: officially they distribute a SNAP and also DEB/RPM packages (I guess the Flatpak is from community), which is what is used for the AUR package (the DEB), which I would use over a SNAP or FLATPAK.
//EDIT2: seems like I have no issue with the AUR version. As I don’t open the keyring when opening my desktop session, it asks for the keyring password when Mailspring starts and everything works. I guess it could be possible to unlock the keyring automatically when the desktop loads (see WIKI), but I’m not gonna test further for me it works with the AUR version.
//EDIT3: seems like I don’t have issue with the Flatpak version either (it asked for my keyring password, I could add a mail account), I will not try but pretty confident the SNAP version would also work properly.
Make sure you have properly installed gnome-keyring, I guess this is your issue. I would still try to remove the Flatpak, and install the AUR version for a more ‘native’ solution instead of using Flatpak.