The problem is that accusing a student of cheating (especially in the US) comes with a risk of lawsuit by parents and being forced to "prove it". If it's not clear-cut plagerism but a new novel work, that claim doesn't hold up.
I think the future of this is milti-fold. Don't try and identify AI papers (as you say, a cat and mouse game), just try to identify if the author is the same author as previous works. Students over their academic careers produce many works, enough to identify them. Secondarily, emphasize more short form writing in class, where research is done ahead of time but papers are composed in real time. Last, especially for younger students, emphasize (and grade) the process, not the outcome as much. Show your sources, outline, etc long before you turn in a paper. The value of having ai write it drops if you have to show your work.
I remember pretty vividly that my ability to formulate good essays varied massively between days. (And it's the same now, though it's code instead of essays nowadays.)
There are days on which I struggle being coherent and others on which things just flow out - sentences chain together into a fantastic essay.
The proposed idea would absolutely fail people like me as cheating.
Honestly, this whole debacle was obviously going to happen and anyone with half a brain predicted it when these services (the fraud detection ones) started to pop up.
I think there's a danger of teachers rating their cheat-detection ability too highly and catching people like myself in the crossfire.
Edit: I see this was not a popular take, but I'm open to hear a practical solution, where teachers don't have to waste time grading the work of an AI instead of actual student work.