Living in Arizona I have the feeling you and I have a very different concept of 'rural'.
I don't think of Arizona when I think rural. To me, rural is where man and nature can exist together and live off the land. There's not much "living off the land" to be done in arid deserts. Sure, it can be done but it will never be a Garden of Eden. Except maybe the upper levels of northern Arizona where some woodlands and more temperate climate conditions exist. Still too arid for my taste.
I would be you would be more likely to run into a bear, deer, moose, coyote, mountain lion, or wolf in CT than in AZ.