The reason many apps stop showing ADs when their internet is blocked is because they need to make an API call to their own servers before running the AD. That is the common behavior but not mandatory
1. App can't use mobile data in background
2. App can use mobile data in background except in Data Saver mode
3. App can use mobile data in background regardless of Data Saver mode
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For anyone doing comparisons, the literal settings appear under "Mobile Data Usage" as:
* [X] Background Data ("Enable usage of mobile data in the background")
* [ ] Unrestricted mobile data usage ("Allow unrestricted mobile data access when Data Saver is on")
Confirmed these settings on One+15 on OOS16 (based on Android 16).
Is it also the case for other Android brands?
P.S. I did use it before to turn off ads.
Some Chinese/Taiwanese brands do it too, but most western brands don't seem to include a firewall.
None of the Samsungs I have owned so far had this feature and neither did my last Pixel.
There are also (open source) firewall apps that will let you block (non-system) apps if you're on a stock ROM like me.
Technically, this is a permission, just not a user-grantable one. Google has moved quite a few permissions from inherent to user-grantable, but most apps don't work without internet (unfortunately) so I doubt they will do it for the internet permission in stock android.