re: Also, a balloon wouldn't work
"It's too windy on top of Mt. Everest for a balloon or blimp."
You would need to use balloon on less windy days. But you aren't going to try to land balloon on the mountain. A winch on ground at some location which is at +5000' elevation can play out balloon with cable. Another winch [5 ton] on balloon. From Balloon winch cable line hang payload, say 2000' below balloon. Connect to payload a additional cable which is used to direct payload to mountain top. All this could evolve into tram system. An intermediary stage could be tram line to direct payload to mountain top using balloon to lift the weight.