Those places are all over the US but just not very interesting. Ft. Davis (or maybe Alpine) Texas is what you're describing but no one really goes there because it's just a small town with people working and living their lives as best they can. There's nothing to attract any outside attention really. I only know that area because of my wife introducing me to Marfa TX which does a little bit of tourism because of the Judd foundation.

I fell in love with that part of Texas and the people. My wife and I were married in Marathon which is in a ~75 mile radius of the towns i listed above