It doesn't need it, but it does allow it to be more like the Play Store. No need to download then tap install which pops an Android prompt to allow install/update nor any need to allow from unknown sources in settings.
With the privileged extension it's exactly like the Play Store: you tap install and it downloads, installs and updates the apps in the background for you without any prompts. It's technically possible unrooted with some adb hacks, but the privileged extension is the technically proper way to be a store. Without it, it needs that user interaction with the app install popup window to let it through. That's not F-Droid being nice and confirming, that's enforced by Android.
In the context of the article, allowing the user to allow this for any store app, puts every other store on exactly the same ground as Google. The Play Store is not special in any way other than that it has that special store app permission that can only be granted via an XML file on the system partition.