You can't build infrastructure where you can't reach with construction equipment
High Mountains like Mt Everest get thick and unstable icecaps. Thier geography is significantly changed from month to month by large landslides and snow storms that make building any sort of traditional Road or Railway impossible. Cable cars are generally preferred for traversing mountains over trains because they have much smaller footprints on the mountain. You dig down through the snow and ice, anchor the piles deep and minimize thier profile in hopes that they can survive the landslides.
Larger mountains tend to have more unstable icecaps; so, I'd imagine that even if you could install a cable car system that high up, that the landscape would be to unstable, even for cable cars to be a safe option.
But the real restriction is going to be installing the piles at your higher altitudes. No ground vehicle can drive up Mount Everest, and heavy lift helicopters can't take off and land at altitudes above 10,000ft; so, while you can daisy chain cable cars up the first 1/3rd of the mountain, getting the heavy equipment you need to the higher parts to continue installing piles will be impossible without a heavy-lift VTOL aircraft that can reach the summit... and if you have one of those, you don't need an expensive train or elevator.
For reference, the highest cable car in the world only reaches 14,783 ft above sea level, about 1/2 of the OP's needed height, and this cable car is a bit of an exception because Tianmen Mountain is in a particularly warm and arid place allowing it's icecaps to fully melt in the summer making it accessible to construction vehicles in the summer. The highest mountain that does not have permanent ice caps it Monte Pissis at 22,283 feet. So, at most you could maybe get about 3/4 of the way up your mountain by cable car IF it is in an absolutely ideal location unless your world is significantly warmer or dryer than Earth.
The other consideration is that your construction crews need to be able to survive in the environment. A healthy person will typically pass out at a 43% reduction in atmosphere and die at a 73% reduction making the summit of Everest at a 66% reduction so hazardous that only top athletes can stay conscious up there, even at rest. However, construction crews are often not top athletes and they exert themselves in short burst of high energy which is much more dangerous in a reduced atmosphere. OSHA regulations say that anything more than a 7% drop in oxygen is potentially dangerous for construction crews to work in; so, even if you could deliver the needed equipment to the top of the mountain, present day robotics are not advanced enough to do this without human labor and human labor can not function at nearly these altitudes.
A subway works in theory, but may be a poor choice for economic reasons
Since you can not build over the mountain, you might be able to build through it. If you build a system of pressurized subway tunnels you can go under all the unstable snow and reduced atmosphere, and simply drive your heavy equipment up the tunnel as you build it. If you spiral a tunnel up at a 10 degree gradient, then you would need a 31.665mi (50.96km) long track. While this might be doable in theory, it's probably not doable in practice. Subways cost about 2.5 billion dollars per mile of track. In major cities these costs can be recovered by charging millions of people daily commuter fees, but for a remote train system that most people will never use, and no one will ever use more than once, the total price tag of nearly 80 billion dollars will likely never find an investor if alternative ways to the top exist.
A Helicopter can make it if you find an updraft
On May 14, 2005, a Eurocopter AS350 Squirrel helicopter became the first to land on Mount Everest's Summit. There is nothing special about the helicopter itself, but it does require a special pilot. Although the altitude required is on paper too high for any modern helicopters to climb, mountains are often hotbeds of updraft activity which can be rode up to altitudes higher than a helicopter's theoretical ceiling.
Once you get to the top, ground pressure will help you take off just enough to clear the summit and begin your controlled fall until you reach pressures height enough for sustained flight. You will need a better helicopter than an AS350 to carry any passengers, better options do exist that should cost less than 10 million dollars a helicopter.
Hybrid Airships
There is one kind of aircraft that you forgot to consider that can hover and land at much higher altitudes than a helicopter: Airships. While most airships today are either not designed go above 10,000ft or are not meant to carry passengers, this is more of an issue of environmental controls than of any actual technological limitation. Lockheed Martin’s High Altitude Airship program developed a hybrid airship that is designed to operate uncrewed with a 4 ton payload at 65,000ft or with a crew at 10,000ft with a 23.5 ton payload. Presumably, it could get to the top of Mount Everest with about a 10 ton payload which is probably enough that you could find a way to bring up your heavy construction equipment to build a cable car system, but still does not address that the altitudes are too hazardous to work in. That said, it is enough to be able to ferry many passengers a day up to the top of the mountain and back, and it only costs 40 million dollars; so, you could operate a fleet of hundreds of airships for cheaper than you could build a practical train system, or 1 airship that can carry dozens of passengers with 1 pilot for the cost of 10ish single passenger helicopters trying to do insanely dangerous maneuvers to ride the updrafts up making it perhaps the most economically viable solution.
That said, there are different optimal time windows for when each kind of aircraft would be ideal. If you assume your mountain has seasonal winds similar to Everest, then mid-spring and mid-fall would be the best times to operate airships when the winds are their weakest, but as the winds get a bit stronger, helicopters might be better for riding the stronger updrafts with mid summer and winter being complete no-go times. So just depending on how many people you have visiting this mountain and how important seasonal restrictions are, will determine if you should opt for subways, helicopters, or hybrid airships.
