Golf in the mountains can be as much fun as golf anywhere. Having said that, the terrain and altitude are two problems that are difficult to overcome.
Ideally, a mountain course would be on relatively flat land (< 200 ft of elevation change on site) and surrounded by spectacular mountains. There is not a lot of this type of land that could accommodate a golf course; additionally, a course needs to compete with other land uses.
Extreme land can lend itself to some incredibly fun shots, but usually the overall qulaity of the course suffers.
The altitude question is one that I think gets frequently talked about, but unless someone has a lot of experience golfing at altitude (> 6000 ft), it is hard to fully appreciate.
Disclaimer- although I live in NC, I grew up out west and wouldn't really call the Apps mountians. Nevertheless, Wade Hampton and the Homestead are wonderful courses, but being only at three to four thousand feet, the effects of altitude are not that great.
Obviously the ball goes farther, but that also depends on the trajectory of the shot. A high ball flight will benefit more from thinner than lower ball flight, even if those shots travel the same distance at sea level.
Furthermore, as Bill points out, the ball does not curve as much, which by itself detracts from the game.
In some ways, golf at altitude suffers from the same negative effects that technology has placed on the modern game. Twenty years ago, it was more of a finesse game where one had to control trajectory and shape. Today, it is blast it high and the ball goes straight- very similar to the nature of the game in the mountains.
SP