Alternative app that's not encumbered by the Play Store's rules: https://f-droid.org/en/packages/com.ghostsq.commander/
Its functionality is hindered by Google in that it can not download and install updates in background as Google Play does. One has to root the phone and loose access to banking and SSO apps to get enough permissions for F-Droid. EU anti-competition monkeys should have looked at that a long time ago.
Given that trying to update a package from the notification just yields an error and this issue hasn't been fixed for 5 years (https://gitlab.com/fdroid/fdroidclient/-/issues/669) and the F-Droid repository is known to update so slowly that most devs recommend adding their direct channel, F-Droid doesn't really feel like a usable alternative.
I think unfortunately the best bet here is to wait for some commercial player to step in and make a decent alternative.