We already have such tools. This is an example: http://fotoforensics.com/

An issue that the same algorithms that can be used to differentiate between real and fake can be used to train AIs to make better fakes. But here, the fake detector has the advantage in the arms race.

Another approach is to use cryptographic signatures. Some cameras sign can sign pictures to ensure they are original.

I can point the camera at a high-resolution screen or printed out photo though.

I don't think it would work. A 4k screen has 8Mp, that's less than a cell phone camera. The artefacts will likely be obvious to those who know where to looks.

Printing may be a better option with regard to resolution but because the print color gamut is different, simple analysis can probably give away the trick really easily.

Furthermore, assuming EXIF informations are signed along with the rest of the picture, you also make all the metadata match the picture. A sharp picture and bright daylight is not consistant with a slow shutter speed for instance.

It will bring the balance back to privacy.