I've used the 4PA for a long time with good success. White sacrifices time in exchange for space, a very dynamic idea. In its cxd5 form, it's underrated. The main idea is to strive for e4-e5 and then ram the d-pawn down Black's throat (if Black fails to exchange on e5, then e5-e6 is usually strong). There is a book, some years old now, by French GM Anatoly Vaisser, which is a must-have for 4PA players. Vaisser has played the 4PA innumerable times, and his games merit study. For a long time in the 90s, Black's rock-solid rejoinder was supposed to be 6...c5 7. d5 Bg4 8. Be2 Nbd7, but after some years, it became apparent that this does not stop White's play if he hurries his h-rook to e1. Nowadays many Blacks rely on 6...Na6, but as discussed by Vaisser in the latest NIC Yearbook, cracks are also appearing in this system. I think Black's best play is, in fact, the theoretical main line: 6...c5 7. d5 e6 8. Be2 exd5 9. cxd5 Re8 (as an example of the juicy possibilities, if 9...a6? then 10. e5 is very strong, for example, 10...Ng5 11. e6 fxe6 12. Ng5), but this requires thorough preparation and gives White exactly the sort of game he wants. One word of caution about this line: don't play d5-d6 before Black develops his QN, otherwise, it'll jump out on c6.
The system with exd5 is crushing if Black doesn't understand that he must prevent f4-f5. But if Black knows what he's doing, it's pretty harmless. I don't care at all for the lines where White meets c7-c5 with something other than d4-d5. Black exchanges on d4, and White has a very loose sort of Maroczy Bind.
If Black plays 6...e5, then 7. fxe5 dxe5 8. d5 and the game is less open, but the exchange of his f-pawn for Black's d-pawn is very good for White, who will often strive for c4-c5.
I don't say the 4PA beats the KID; I merely say it gives White reasonable play for the win, and would be a good weapon for anyone who enjoys open positions.