That makes sense, if archiving saves space then it’s just removing/uninstalling the app, but leaving the icon behind. Since you’ve manually installed the app from obtainium, it can’t do that because it has no idea how to re-install the app when you need it again.

If I had to take a guess (which is exactly what this is, a guess) it is because Android doesn’t “know” where the app is from. I assume the Play Store has specific (system-level) APIs that it uses to “tell” Android how it can be restored (or rather, Android can signal to the Play Store to do a reinstall) when you go to unarchive the app.

It’s been a while since I kept up with the latest in Android’s APIs, I’d heard there were some APIs that third party stores could use to be recognized as a store, but I’m not sure what the requirements are for that (such as being a system app rather than a user app, or signed by the ROM’s keys) and if so, whether archiving even supports third party stores.

I can’t think of any other workarounds unfortunately, especially if you want to persist app data. Perhaps there’s an app that can make custom widgets that look like an app entry on the home screen (and allows setting an icon/text) but I’m not aware of any, and that definitely wouldn’t save the app data.