like a button

No. Then when Dumbledore waves his wand it goes bing-bing-bing when he gets to the button. With ygert's theory, it goes bing-bing-bing when he gets to the ring, Harry detransfigures the stone, and then (with non-negligible probability, at least) Dumbledore doesn't think to re-check the ring itself.

I don't find the theory convincing, but if Harry were going to do something along those lines then using the ring seems smarter than using the button. (Better, for topological reasons, to transfigure Hermione's corpse into part of the ring. Otherwise, what happens if Harry's asleep wearing the ring and someone casts Finite Incantatem? Eww.)