Engines are not my forte. But, I know just enough to answer your question. There are many stages in which a modern aircraft engine is tested. What you are suggesting is called a static or bench test. An engine is put on a test stand, and a "bell mouth" is added as an air intake. The engine is then tested for various power settings, environmental conditions etc. GTRE, HAL, NAL etc. have these test beds in Bangalore. Your question is, why not make one on top of a tall mountain.
This is not feasible for two reasons. Take Siachen for example, just maintaining food supplies for soldiers is a pain. And this is at 20,000 odd feet. Chinooks can barely lift its own weight at that altitude. Imagine the complexities of maintaining a jet test center there. Two, planes fly higher, much higher.
But then how about something more realistic, say Leh? While it is possible, it is not needed. Static pressure and temperature conditions can be emulated in labs even at sea level. The question is whether you are ready to spend the money to build such a facility. The West and Russians did it half a century back. But, we said, building test engines takes away funds from growing enough rice and bananas. Which is okay, but what is holding us back now? God knows.
Thankfully, as part of the C-17 purchase, we are getting an old derelict HAT facility to GTRE. At least, we can start, i.e. if the said facility can be kept operational. I have heard things that it is hard. Getting it here is fine. Keeping it running requires parts and spares which requires clearances and further imports, and further mullah.
After the static tests, there are wing tests, like what GTRE did on the flying test bed in Russia. We need one of those in India, if we are anywhere serious in designing our own engines. After that, in case of a fighter jet, it has to coupled to the inlet. An ideal inlet would modify the airflow entering the inlet to a steady airstream which provides good engine performance. However, an ideal inlet is very difficult to design. CFDs are getting there in simulating the airflow inlets, but it is not there yet. Till then there is no alternative, but to do actual testing.