Don’t you think app developers should have the freedom of choice for app stores? Android has multiple places to get apps, lots of FOSS apps aren’t on play store, and likely aren’t on iOS at all. Having the option for 3rd party store gives not just users, but developers too, an option and access to more apps/users that wouldn’t otherwise be available. You want safe secure apps, that’s all good.
You mentioned that an app might move away from the app store to a 3rd party store, if you already use and trust that developer’s product how will having to get it from a different location change that?
Now, not to sound too harsh, but why your convenience of not having to go get another store more important than a devs ability to host their product in a different location?
Is battery optimisation turned on for the app? For me with some apps if it is on it will run in the background but i guess pause it until I actually open the app again.