Side loading the Fly app on to the SC won’t work either, as you’ve seen, it gets hidden.
If your SC is rooted you can unhide it using:
su -c "pm enable dji.pilot.pad"
(replace pilot.pad with the Fly package name)
But even then, I’m still not sure the “regular” Fly app would work that way as the version shipped in the SC firmware update could be different.
Just checking, after installing the latest SC firmware you did then update Fly to v1.4.4 too? As the SC firmware still currently ships with the older v1.4.2