Mountains are objectively aesthetically pleasing to humans. They also retain the most wild and specific nature around, especially in western world where in the past mostly agriculture changed environment forever. You can find entire states in Europe where 'wilderness' is basically a well curated park that was a field some time ago. Air is much cleaner, you see far and you feel utterly insignificant up there.
Mountains also allow tons of various sports most of the year as you mention and attract such crowds.
Overall, people go in droves to mountains every year, summer and winter, to recharge. And its a different recharge compared to some beach holiday, much more active and I dare to say a more effective one (active vacations simply recharge better mental tiredness of modern living and working).
I live near Geneva, Switzerland and after moving here basically became mountain sports addict. Hiking, ferratas, climbing, a bit of alpinism in non-winter, and skiing / ski touring in winter. For exotic places there is diving to keep a bit of balance. Did also some paragliding too but had a nasty accident last year that almost killed me so stopped that. Rest I continue, and will do till the last moment I can still do them, they make me properly happy like a toddler. I live in best possible place in the world for people like me, 1h drive from home and I am right below Mont Blanc, towering almost 4km higher than Chamonix valley.
One drawback as you mention - higher mountains are basically high altitude deserts, snow and ice everywhere but humidity easily below 5% ie in Himalaya. Many folks trying ie Everest didnt get higher than base camp simply due to being often sick up there, amateurs and pros alike.