If you check out the wiki article then you can see that you are absolutely correct. You can get a system like this working on a mountainside as long as it faces the sun. I think there are some engineering tradeoffs that have to be considered and also environmental concerns. I think it's a lot easier to find a desert with minimal ecosystem compared to a mountainside

unless you had a perfectly vertical mountain your tower would end up being longer and costing more. if you did have a perfectly vertical mountain it would still be somewhat difficult to build on the side of a cliff.

I suppose you'd need a mountain in the desert as well.

The mountain would also provide unwanted shade.

Not necessarily. Depends on directional facing. A south facing spur up the mountain may provide no shade at all to the base.

And longer, yes, but not in need of vertical support.