I've started developing something like that, and I started by noticing like you that basing on screentime isn't fair to apps that are purely in background, so I tried to compensate for that.

Then I found that some apps, even though they were useful, didn't actually do much under the hood. Like say Frost for Facebook which is mostly a webview with few extensions. So I wanted to add a section for things that are mostly a "viewer". An additional example of such a thing is a media player: if you look 10h a day at videos, it's thanks to the content, not to the viewer.

And then, there are behaviours you maybe don't want to trigger, like getting people addicted. 2048 using the screentime metric would have gained a lot of money (and it does deserve a lot!), but IMO people who got addicted to it should give them less.

And what about apps that have paying variant on Google Play? Which deserves more money, the app that is simply free on fdroid and Play Store, or the app that is free on fdroid, but paying (or free reduced) on Play Store.

So, I spent few days trying to make something fair, and at the moment I believe that the most fair I can do is to give equally among all apps that I've used in the last 7 days.