The first section of this answer is more subjective but I hope still insightful.
The KID isn't necessarily the best choice for a must-win game but it is a good choice for a fighting game and for creating chances to win as black. This is because it often creates a game where White will need to make many difficult choices and there are chances for White to go wrong.
It's advantage over other openings is that it is more strategically complex and has less chances for simplification than solid mainlines. Yet is still more sound than extremely dubious lines such as the Budapest or Albin.
Now for a more objective analysis of the position you provided:
It is true that White has traded queens, but he has also traded his strong d4 pawn for Blacks d6 pawn. In this position Black is actually strategically better, the d4 square is forever weak and White is potentially overextended on the queenside. While Black can easily cover the d5 square with c7-c6. However it isn't so simple because black is also behind a lot in development.
So we can understand that the position has become a dynamic fight between Whites initiative and development vs Blacks strategic advantage and control of the dark squares.
So to conclude: Yes the statistical draw rate is high, but Black is still getting a fight and it isn't easy for White. In general playing for win with black against a strong opponent is the most difficult task in chess and there is no perfect solution.
Edit: I will also add that the exchange variation itself has it's own body of theory which White needs to know before playing it. Most players with white aren't looking to draw (not in a must-not-lose situation) and so don't play the exhange variation regularly. Whereas Black KID players will be already familair with the theory of the exhange variation as they encounter it more often.