According to https://www.xda-developers.com/android-12-alternative-app-stores-update-apps-background/ https://www.xda-developers.com/android-12-alternative-app-

According to https://www.xda-developers.com/android-12-alternative-app-stores-update-apps-background/ https://www.xda-developers.com/android-12-alternative-app-st..., it could do it, it's just not implemented. 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 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.