Android 12 has a mechanism to allow an app that installed an application to update it in the background, but the client needs to be updated to support it.
F-Droid hasn't yet, see issue here [1] - some of the other F-Droid clients, like Droid-ify have [2].
Exactly, this is only tangentially related to rooting. Google doesn't need root on your device for their closed Play Services to install software, but the component that you want to have this installation capability does need some system-level permission. Many people grant it that by rooting the device, but installing something like /e/OS (=Android with microG and a few other improvements) is also a way to do this.
SkyDroid can update apps without user interaction (even on non-rooted devices) using a workaround which requires a one-time ADB setup. You however still need to open SkyDroid and click a button to start the mass-update process, but this is an intentional design decision - it makes sense to check which app updates are available before blindly updating everything.
