From what I learned, only apps that target Android 12and above could have this background update feature. I'm on Android 13 but none of the apps I use updated. I had to switch back to using the f-droid privileged extension.
I'm on Android 13 currently, I don't have root, stock OS. All my apps update, including ones that are marked in Fdroid as "created for older Android releases". I use around 15 apps from that source and none caused me much trouble. A few times had to invoke installation manually, but honestly I think it would update automatically with more patience on my side.
That's why I'm so pumped about Fdroid Basic - it's a much better experience.