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.
A nice unexpected benefit has been that Insular does background installations and updates of APKs. When I click "update" in f-droid, I now choose Insular as the handler and it happens in the background. It's much more reliable than however f-droid does it internally.
I suspect that some banks use a whitelist rather than a blacklist for certain phones.
Tricking out these detections is a nightmare.
Sorry if I'm appearing snarky but that's the whole point of this thread. Google obstructs third-party app stores.
https://f-droid.org/en/packages/org.fdroid.fdroid.privileged...
(Not saying it's appropriate to impose the "devs must update apps for each Android version" requirement this way.)
Fdroid is very close to being the "brew" or "apt" of the Android world, being able to pull updates from multiple sources.