You are getting lot's of good advice, but somewhat conflicting messages.
I have and own horses and elk hunt. I wasn't raised with them, but my wife has a passion for trail riding(not hunting), so I use them to pack meat.
I think unless you have a passion for horses, you won't end up owning them for long. They are a huge commitment. If you have a knack for understanding them, that will make all the difference, but as others have said, if you own them you need to keep them in shape, and this means both physically and mentally.
A few things you should know. Horses are lazy. All they want to do is hang out with their friends, of which you are not one, and eat. So you need to get them used to working on a regular basis, so that means riding them a few times a week and doing so in a manner that will prepare them for the work you intend for them while hunting, so a casual 3 mi ride on a flat trail won't cut it. Also, getting them into scary country where noises and smells exist so they and you can get used to handling that stuff. This takes a lot of time and money, and takes away from your own time to condition and scout.
Horses are afraid of everything. The only thing that makes them okay is getting them use to everything. A horse that doesn't mind a pack, doesn't mind the smell of blood/meat, crossing water, or riding in the dark may one day decide that some gray log is bent on killing him. Trust me this is no joke. It clearly helps if you are capable of being calm and cool in all situations as the horses clearly feed off of your confidence or fear. But, no matter how good the horse you will need that confidence at some point as you will have something happen.
I have never rented horses. I will say that it would be good to have a horse that has been worked regularly and packed in the woods often, so a rented horse might benefit from that. The downside I see is that each horse tends to have it's particular strengths and weaknesses, such as blood, water, hopping logs, buddy sour, etc. Obviously when renting it is unlikely you will have the benefit of that kind of understanding.
The biggest hassles in having your horses around while hunting is having to feed and water them. They require a lot of food daily, particularly if they are working hard. So much that they can eat their body weight in a week's hunt. As mentioned earlier, they need to water regularly, though often won't want to drink at all the first day which can subject them to Colic and kill them. What this means is that you will have to be very particular on where you camp and how to manage to get feed for them. Remember this when hoping to get 20 mi from the trailhead, especially if hoping to get near the top of the mtn.
Horses weigh 1000-1200lbs. Ounce for ounce they are 2x as strong. This means they are easily 10x stronger than most men. Horses are PREY animals, so like an elk when confronted with danger they will turn and run NO matter what is in the way(you, cliff, etc.) and mind what I say here, are the most likely way you will die or be crippled while hunting.
So, what I do is take as much of the risk out of the equation that I can. I don't camp with them, I don't ride them while hunting. If I kill an elk, I go get them and pack it out. See hiring packer as alternative here!
Good luck.