Beyond citizenship, essentially a Jew can't travel there. As I understand it, if you answer Jewish to the religion question on the Visa ap, you'll be denied. I believe non-Jews can travel only with an approved sponsor or on an approved tour, you can't just travel around Saudi Arabia.
That part of your post isn't exactly accurate. I was in the kindom 1980-1981. You could travel freely, unaccompanied, although it may not be considered the wisest course of action, due simply to the environment. Just prior to my arrival, an American couple had gone out to the desert outside of Riaydh, and became stranded for a day or two. They were lucky, as they were searched for, and found in time.
The desert of the Arabian peninsula is a very harsh, unforgiving environment, and must be respected, or the ultimaye price paid...
the infowarrior