Isn't 1. d4 Nf6 2. c4 d6 (Old Indian Defense) a more flexible move-order than 1. d4 Nf6 2. c4 g6 (King's Indian Defense)?
I don't see why any Black player who wants to play the King's Indian would avoid the former move order.
Playing 2...d6 allows for the Old Indian Defence and King's Indian Defence, while 2...g6 allows for the Grunfeld and King's Indian Defence. Both have some degree of flexibility, so it depends which openings you are fine with playing.
However, against either 2...d6 or 2...g6, White can play 3.Nc3 (threatening e4). At that point, you would have to choose what opening you're doing to play (so the flexibility ends here). In the 2...d6 3.Nc3 position, 3...e5 leads to the Old Indian and not playing it will lead to a KID (or KID-like structure). In the 2...g6 3.Nc3 position, 3...d5 leads to the Grunfeld and not playing it leads to the KID.
The main reason to use one move order or the other is if you know your opponent doesn't like facing one of the two openings you can deploy. For example, if your opponent likes to go into the Classical King's Indian with Nf3 but detests facing the main line Grunfeld, he might play 3.f3 after 2...g6. Then, you can revert to the KID and your opponent is stuck in the Samisch. Comparatively, if you had played 2...d6, your opponent would go 3.Nc3, 4.e4, 5.Nf3 (since he doesn't fear a Grunfeld happening), reaching a variation of the KID he likes.