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.

On my Moto FDroid does a decent job of keeping my apps updated. I still have to intervene about half of the time though.