Why do India and China keep fighting over this desolate terrain?

Because if a country is not willing to fight to control it’s territory anther country will claim it and allowing another country to claim your territory will lead to more claims and gradual erosion of territory, potentially escalating to all out war and attempt at total invasion and subjugation.

Controlling the mountains gives you a huge military advantage. You can defend much more easily if you control mountains. You can attack into non-mountainous terrain much more easily. You do not want to be on a plain exchanging artillery fire with troops dug into mountains.

Given the recent conflicts between India and China (like the 1962 war) only a fool would give up such terrain.

I wonder if armed flying drones will change that. Could a very fast, agile, and silent autonomous drone engage targets in mountains as easily as the do in other terrains?

Geo-political mistake by China, that land is not worth having India against you.

I expect it's water. Aren't there springs for major rivers?