While the KID is a good opening if you want an exciting game and good winning chances as black, these benefits come at a cost. The KID is not a solid opening when compared to many other ways for black to meet queenside pawn openings from white, and there is a wide variety of tricky lines that white can choose from to try for a serious advantage out of the opening.
Some of these setups by white are very sharp, forcing black to know a huge amount of concrete lines by heart not to immediately fall into a lost position (I'm thinking mainly of the various ways to play the Mar del Plata variation for white, including - but not limited to - Kramnik's dreaded bayonet attack)
Other setups are more strategic in nature, trying to utilize the space advantage that black hands over in the beginning of the game. In this list, we include the Sämisch, the Petrosian, and the Makogonov systems (not to mention the various fianchetto systems that white can try, which is another can of worms that black has to deal with). All of these systems pose serious strategic problems for the black side, and black needs not only a solid understanding of the arising position, but must also know a ton of spectacular "only moves" to defend properly (see the game between Svidler and Griscuk from the candidates 2013; if Grischuk hadn't played the seemingly outrageous 12...Nxc4! he would have found himself in deep, deep trouble).
If you want a good indication of how much effort and time a top level player would need to put in to play the KID consistently with confidence, you may want to look at the book series "Kotronias on the King's Indian". In this series, Kotronias needs more than 2000 pages of concrete analysis to demonstrate that there is no clear refutation to the KID.
With all this being said, the KID is certainly not a bad opening (a claim supported by the analysis done by Kotronias). But the problem with playing it at the highest level is that black is risking a lot when playing it, and would be risking even more by including it in their main repertoire (leaving themselves open to some seriously nasty preps by their world-class opponents; this is something that MVL does by only playing his absolute pet Najdorf and Grünfeld against basically anything, and sometimes he suffers for it). Therefore, most players at the top level who are comfortable playing the KID only use it as a surprise weapon (or if they're playing shorter time controls), or if they're in a must-win situation where the difference between a draw and a loss is irrelevant.