They don't seem like they have background installation mechanism right?

I went all of the troubles creating an automation with Tasker because I haven't found one that worked like Google PlayStore's auto-update back then.

I wonder if there is a couple of them out there that already has those? Maybe like with shizuku, ADBwidi, or root access?

Edit: typo

There are two options that makes me a bit confused.

Auto update apps. Try to install updates automatically

Notify about new versions of applications. Show a notification when new versions are available

Do I have to turn off the later to let the auto-update runs fully in background or It's fine to keep both enabled?

On the first run, I think I had both enabled and I still tapped multiple times. Shizuku installer worked great though.