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Others have covered the fact it's because of air pressure but haven't fully answered why that is the way it is.
It's simple really.
The force of gravity is also at play. As you go higher up, gravity gets weaker as you get farther from the earth's centre.
And it is that gravitational force that increases the air's density, same reason why if you keep going down in the water, the water gets denser.
For the heat to move around you need to be in a sort of goldilocks zone of density.
It needs to be dense enough that the fluid molecules can move around and spread the convection energy around... but not so dense they can't move much either.
Furthermore there's actually a couple different layers of our atmosphere.
First at our level is the troposphere, where heat is absorbed into the ground itself and radiated back out, as well as the perpetual heat from the earth's core, and reflected off the ground too (visible light).
The troposphere is warm and gets colder as you get farther away from the earth's surface, naturally. That heat is absorbed by the air itself so, as you get farther away it gets colder as it has more air to travel through.
Up higher is the Stratosphere, where it's ice cold and the air thins out.
However we get a sudden uptick in temp as we go even higher into what is called the Stratopause, back to briefly warm temperatures between the Stratosphere and the Mesosohere. Why? How?
Simple, this is the little sweet spot Ozone molecules hang out, forming a protective convenient bubble around the earth. Ozone absorbs Ultraviolet light from the sun and turns out that stuff is HOT, so there's a band of a hot zone right above and below the Ozone layer. Think of it as a toasty little bubble around us.
Above is the mesosphere which cools off again and gets back to being really frosty quickly, for the same reason the Stratosphere did, distance.
Then we hit the mesosphere, which is effectively the point when the atmosphere is so thin it stops protecting and is the "outside" of our protective blanket.
You can imagine this like earth being wrapped in a blanket, and the mesosphere is everything outside the blanket. Without any protection you are subject to the unbridled radiation of the sun which means you go back to being really toasty, as you get a bit higher you are effectively in space now and will soon enough hit temps that just cook you alive in a minute or two. Really bad sunburn zone.
So to answer the question overall:
Hot air rises... but only when there is air to rise.
Top of the mountains just don't have enough air anymore for it to really rise much more. It still does but the hot air rising effect just gets weaker and weaker as the air gets thinner due to less gravity.
The strength of the gravity field at the ISS orbit (400km) is still 90% of what it is at sea level. The air thins out at high altitudes because there's less air above it pushing down, not because of weaker gravity.
To add to this, the force of gravity at the top of Mt Everest is about 99.7% as strong as sea level. So you're right that it's not about the strength of gravity itself at that particular point, but about the weight of all the air above that point.
I never heard it explained that way. What an excellent comment. Thank you for taking the time.
It's mostly true, but the basic premise is not. Gravity is not significantly lower in the upper atmosphere.