I feel like this idea comes up often and in my opinion it doesn't solve anything. Take a picture of an AI image and you've made this approach useless. Which then goes to the argument of "well you'll see it's a picture of a picture" to which I will say there are plenty of ways to make this not appear so, and the ultimate form of this argument is that you can eventually project light directly into the photosensors, or otherwise hack the input between the photosensors and the rest of whatever digital magic that turns light into a JPG on your phone.
SynthID survives basic transforms including screenshots/photos, although it can of course be defeated. Even still it helps with the laziest fakes, which there seem to be a lot of - I've seen several quite widespread misinformative images over the past couple months that failed a synthID check.
Anyways I think approaching the problem from both directions is probably good.