It could. But the root problem is, there is no such thing as unqualified "sensitive information". The information is sensitive to someone, for some reason. The problem with screenshots manifests when the app and the user disagrees about whether the information is sensitive and who has the right to control it.

The immediate technical problem is that OSes allow apps to declare what is "sensitive", and then follow those declarations unquestionably, preventing any preservation of that information.

The underlying social / political problem is that platform vendors allow app vendors to unilaterally declare what is and isn't sensitive, and proxy their opinion without question, and with no consideration for users and their context.