Actually the King's Indian has seen a recent rise in popularity. For a few years now, players like Nakamura and Radjabov have revitalised the opening. And if you take a look at the very recent top level games it is striking how many top players gave it a try: Nakamura, Jobava, Grischuk, Giri, Radjabov, Kasimdzhanov, Bacrot and quite consistently Ding Liren, just to name a few.
Those are all games from 2015 and the list reads like a who's who in top level chess. So there certainly is no big theoretical issue, that would dissuade black from playing the King's Indian.
One of the hottest lines right now is the 6.h3 line (white's choice in most of the games I linked), apparently Aronian introduced some tricky ideas for white in this variation (according to Seirawan).
I have heard this Kramnik-Bayonet-story before, but I'm not sure how close it is to the truth. Kasparov himself said, that he stopped playing the King's Indian because it became too much work to keep up with two ultra-sharp black defences (The other being the Najdorf). A few years ago the Bayonet-variation was still widely played (Nakamura scored some impressive wins against it), but for now the focus seems to have shifted to 6.h3.