No, quines are not played-out
I believe there will always be new ways of formulating good challenges, and some of those yet to be made good challenges will involve quines. Saying "we've asked every quine problem there could ever be" is like saying we've thought of every math problem there could ever be.
If you only mean that the current state of quine challenges is suffering, that may well be. It is difficult to think of totally fresh ideas involving quines which are somewhat static in nature (and prone to loophole gray areas). But I wouldn't discount the topic in the grand scheme of things.
Additional note: It seems likely that this recent badly received question played a part in your quine skepticism. I think the underhanded part was its real downfall. Writing good underhanded challenges is arguably much more difficult than writing good quine challenges, since their nature is much more subjective and even more prone to loophole gray areas.