Yeah, but in Southern AZ I use to always worry about finding water for my horse. I could get enough out of cactus for me if I had to, but not for my mount (and at over 8 lbs / gal there's no way to carry enough with you for the horse). Ft Huachuca was "20 miles from water and 2 feet from hell".
In the east water is pretty easy to find. More often it's a pain having to ride to a bridge to get across, but the water is there. I remember crossing (ok, the horse stepped over it) the San Pedro River outside Sierra Vista that out east we'd call a creek . More grazing along the way too in the east. Most of the area I rode in AZ always had a bumber crop of rocks, gravel and sand so any weeds (if you find some) were quickly devoured.
Lot of room to ride, but easy to end up a casualty if you're not very careful from the lack of water and the search for grass.
I remember picking up a date once who told me I'd know her house because it was the only one with grass in the yard (and she was right).
Packing it all in on your saddle horse for a multi-day trip is a tall order that requires the sacrifice of quite a few creature comforts for both yourself and the horse. I've done it before, but I don't recommend it. Your idea of packing in 5 miles the first day and stashing feed is a good one. You might even consider packing the horse with saddle paniers and walking that first day, stashing the paniers with your feed and excess camp gear, then riding the other days. By the time you finished the trip, the saddle paniers would be empty and you could roll them up and tie them behind the cantle.