Re: "In the desert" vs. "On the desert"
Hello Stuart, welcome to Using English!
"To live on the desert" might also mean "to find one's means of sustenance in the desert"; but it seems that (as you suggest) the implication in your text is of the desert as a large flat (metaphorically) oceanic environment.
Sometimes you encounter phrases such as "a sea of sand", which embody a similar idea.
It would not surprise me to discover that this was a regional usage, or unremarkable among those who have business with deserts. Maybe one of our US contributors will know.
Not a professional ESL teacher.