Teachers are very good at spotting when a paper was not written by a student. Any student they've worked with and graded works from they can identify. When someone changes tone, pattern, word choice, and grammer all at once? It's usually not because of a flash of academic inspiration.

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.