No inconsistency has been proved. It is simply unknown whether QED or the standard model are consistent or not. Settling this might have important consequences for the methods to squeeze out predictions from QFT, and hence is at least as important as finding a unified theory of the standard model plus gravity - which is unlikely to have any significant experimental consequences, as you say yourself:I also don't think that relativistic QFT is "the final theory". It's not even mathematically completely consistent.
So far, all observations concerning gravity are in accord with GR, but that's likely to be the case, because all we can observe concerning gravity is about its action on macroscopic systems, and there the classical theory is very accurate (in close analogy to our everyday experience that classical electrodynamics/optics is very accurate although here we know QED as the underlying quantum theory). So it's very hard to find specific phenomena where and how GR (or maybe some other classical field theory describing gravitation better, although I don't know of any clear empirical hint that this might be the case) has to be joined with QT.