I think with automatic updates enabled, they seem to install automatically without confirmation. I usually wake up every few days to a notification that a few apps updated overnight.

It might be the F-droid client you are using doesn't support auto-updates?

I've personally switched to using Obtainium

I use droidify and there was a permission it asked me for to be able to install apps in the background. It still sometimes gives me the pop-up but usually just installs the updates after clicking update (I don't like automatic updates).

There are multiple version of f-droid . the basic version doesn't do automatic updates, I think it is meant for maximum compatibility with most devices. The normal version does do automatic updates but requires more recent android version.

I think Fdroid can do this but since you're switching things around I recommend Izzyondroid