Google put limits in Android such that the Play Store is the only app distribution method that can implement background installation of apps, batch installations of apps and automatic upgrading.

F-Droid provides those features only if you're able to root your phone, which manufacturers actively try to stop users from doing. Also, rooting your phone opens it up to vulnerabilities and exploits.

If you have root and/or install an alternative store as a privileged system app, it will be able to install apps in the background just like Play Store.

Possibly.

But there are (a) services I cannot turn off and (b) apps I really need on android.

Perhaps F-Droid would help with (a) but not (b)

SO I plan to have a cheep android for the one app I cannot do without and the Pine phone for everything else...

But time will tell