I have both and use the horse for hunting and the ATV to haul manure and hay.
More and more of Utah is being closed to off road vehicles. You can't even use them to retrieve down game any longer. And the chances of the elk dropping on the dirt road are pretty slim.
But more so than that, my horses are my companions. I ride them year round. And my hunt would be a success even if we didn't score a kill. Just riding them in the mountains, seeing the fall colors is worth it. Like the other post above, my horses stand guard over my tent at night, they sound warnings for camp intruders. They watch the trail while I watch the hill side, Their sense of smell and hearing are better than mine and they often see the game before I do. Their odor mask my human scent better than the exhaust from a wheeler does.
I never shoot off my horse. But your's should be use'd to the sound of a gun going off. Just tie them up somewhere while you shoot some targets, Start a little farther off and move closer as they get used to the bang.
My horses have never had a problem hauling deer or elk. I suspect all the time I spend on them during the summer helps them trust me. prior to the hunt, Load them up with some paniers and put some weight in them ( a couple 50lb bags of feed works well), cover it with a tarp. Let them get used to you tugging and pulling to get something fastened on their back. Then when you load a deer, the only new thing they have to deal with his the smell of gamey blood.
Loading a deer or elk quarters is a job. especially on a steep hill side. It usually takes me and my hunting partner to lift and open the panier bags. Which usually doesn't leave anybody to hold the horse still. Teach them to stand while you load them.
And finally get them in shape. I see so many people bring their pasture potatos up hunting and just like you or me, they huff and puff when they are out of shape. If you wear the horse out he won't want to work for you. It's no fun to see your horse crumple and roll down the mountainside (with your rifle in the scabbard) because you over did him.
Most fall hunts, the grass is gone. Plan on bringing in your own feed. At least some grain for energy. Horses from the flat land sometimes need a blanket. Standing tied to a tree all night at 9000 foot elevation is a lot colder than the night before when they lounged in a barn at 2000 foot elevation
Packing elk out