I think that (the compositor doing the screenshotting) is already the way it works. But "screenshots" (i.e. fed to the user as files / on the pasteboard) are the exception, not the rule, of the use of this API.

The normal, non-edge-case use of the relevant screen-capturing API is for screen-sharing ala Zoom, or for remote desktop using RDP et al. These are use-cases where one app wants to see what's going on in other apps — exactly the thing you wouldn't want a malicious app to be able to do, but also exactly the thing that you do want these apps to be able to do.