Notification Channels is the official way to do this on Android, but it's up to the app developer to categorize them properly. They have no incentive to allow you to turn off ads.

Too bad about the walled garden or you'd have this tweak already installed years ago.

That would be nice. I wouldn’t be surprised, as on-phone models get more capable, if we don’t see them start to build an “inbox” like we see with email where you can then start seeing much more heavy processing happening.