I go quail hunting in the Tonto National Forest north of Scottsdale every year and spend A LOT of time doing it. It really is good enough to travel for from my outskirts of Tulsa location. But, that's in the fall and winter. I wouldn't want to live there in the summer, personally.
My understanding is that AZ deer tags can be kinda hard to get even for residents, and when I found that out, AZ went off my "escape from California radar."
According to my brother in law, Nevada is just as bad now.
If deer and elk hunting were important to me, I wouldn't even think of moving to Arizona, personally.