"Judging by the number of corpses littering Mount Everest, this mountain would also be covered with the bodies of supply runners who didn't make it up/down the mountain."

To be completely fair, you don't need a mountain to be extraordinarily high to be pushing through lower levels of clouds. Really you could see that on peaks ~10,000-18,000 feet, which is a sizeable mountain, but not something that's going to kill people the way the Death Zone on Everest does. There are plenty of ranges in that elevation that do have porters bring supplies up regularly, and no, they don't die at any appreciable rate.

But would it be very inconvenient to maintain some sort of cabin or encampment there? Sure.

"He would also be suffering from oxygen deficiency due to the altitude."

Not really. It certainly would take a few days to fully adjust to that altitude, but you can remain indefinitely.

"His food and water would be frozen solid"- Well, plenty of food doesn't really freeze. Jerky, nuts, crackers, candies, cheese all remains pretty edible. Yes, water freezes, but you can just melt snow

"he can't even light a fire to defrost them, because the other beacon tenders would see it and think that they were supposed to light their own beacons."

Shouldn't be too hard to have a fire built with a bit of protection around it to protect it from wind and visibility. The pyres are about 10 feet tall of firewood- surely people can distinguish between a small, hidden campfire and an enormous signal fire.

Judging by the number of corpses littering Mount Everest, this mountain would also be covered with the bodies of supply runners who didn't make it up/down the mountain.

To be completely fair, you don't need a mountain to be extraordinarily high to be pushing through lower levels of clouds. Really you could see that on peaks ~10,000-18,000 feet, which is a sizeable mountain, but not something that's going to kill people the way the Death Zone on Everest does. There are plenty of ranges in that elevation that do have porters bring supplies up regularly, and no, they don't die at any appreciable rate.

But would it be very inconvenient to maintain some sort of cabin or encampment there? Sure.

He would also be suffering from oxygen deficiency due to the altitude.

Not really. It certainly would take a few days to fully adjust to that altitude, but you can remain indefinitely.

His food and water would be frozen solid

Well, plenty of food doesn't really freeze. Jerky, nuts, crackers, candies, cheese all remains pretty edible. Yes, water freezes, but you can just melt snow

he can't even light a fire to defrost them, because the other beacon tenders would see it and think that they were supposed to light their own beacons.

Shouldn't be too hard to have a fire built with a bit of protection around it to protect it from wind and visibility. The pyres are about 10 feet tall of firewood- surely people can distinguish between a small, hidden campfire and an enormous signal fire.