On a 10 day backpacking hunt in the mountains ounces matter. Even an extra pound will feel 10 times heavier on day 3 after covering thousands and thousands of vertical feet. So I do not think you are wrong in wanting to explore how to handle the awkward weight of a rifle. Even though it is probably anywhere from 6.5-9.5lbs, that weight does make a difference over time as it is just awkward to carry.
Strength is important, but just because you are strong does not mean you will have the conditioning to hunt/carry a rifle in the mountains and just because you are conditioned does not mean you will have the strength to do likewise. I have known powerlifters who have crazy numbers who have tried mountain hunting and after about 1/4 mile up the mountain they collapse in a heap of sweat and sound like they are about to die. I have also known marathon runners who could walk all day uphill, but the second you put 35lbs on their back they collapse like a wet noodle after about 1/4 mile.
I do think you should train the "rifle carry" because it makes sense and there are times, it could and should be used. However, a couple others have already mentioned some alternatives that are the primary way I carry my rifle. Usually I strap my rifle to the sling pocket of my pack and carry it on my back to get into the country I am hunting. I also use that method for a long distance trek on a given day from main camp to spike camp, this could be anywhere from 3-8 miles, if it is further I move my main camp.
If you are hunting the mountains then you will probably at least be peripherally concerned about bears and need to train getting your rifle out quickly. I live in very dense grizzly country and have literally run into hundreds of them while hunting. My hunting partners and I have a battle drill when we encounter a bear. We usually smell the bear before we see it, but once we do identify there is a bear about I grab his rifle off his pack and he grabs mine. This is a drill that did save me/us once. One time we came up over a mountain pass with the wind blowing into our faces and a grizzly that we did not see prior was stalking us, it was about a 100 yds when I yelled at it, it promptly stood up on its hind legs, identified us, and came in at a full charge. At 100 yds you have about 5 seconds before the bear is on you, my friend, upon seeing the bear stand, immediately grabbed my rifle and I his. I fired a warning shot at the bear at about 40yds blowing rocks up in his face and he proceeded to take off up the mountain. I am convinced the bear thought we were caribou, but with the wind blowing so hard in my direction it would not have figured out we were human until it bit once or twice.
The other method I use is the Kifaru gun sling/boot. You can get the universal carrier which allows you to have the rifle on a quick release right at your side. You can have your gun in your hands in about 1.5 seconds. The system takes a little getting used to, but works very well.
As for training.
I train to hunt all year long and have honed my training to match my mountain hunting precisely. As far as rifle weight training. I almost always carry some type of weight in my hands when rucking. The weight could be anything from a sledge hammer, mace club to a kettlebell, but weight is almost always there. It helps you get used to having something awkward or heavy in your hands which not only helps with carrying a rifle when necessary but also translates well into pushing brush out of your way all day.
As for more general variety day training. All backpack sustained hunters, especially mountain hunters, would be best served to work agility training into their variety days. Serious mountain hunting takes a lot of agility and balance. There have been times where I have had to traverse small ledges where if I slipped I would fall hundreds of feet and certainly die. You need to be able to go over obstacles, under obstacles, balance on one foot in precarious positions and climb. You also need to know how to fall well as this could save your life or prevent serious injury. There are lots of ways to train this. Personally, I climb in the mountains and play/wrestle with children/teenagers a lot. Sometimes it is sports or playing different games in the woods. The woods provide a great obstacle course of climbing over and under trees, taking big and small steps etc. If you have a formal obstacle course near where you live go use it. If not, and you live in a city, go to a community park and make your own from the kids playground equipment. I take my kids to the park all the time and we play agility games, it is a fun way to get your training in and spend time with your children.
I will not go into my strength and endurance training, but you should definitely bias your training toward moderate strength and the ability to do "work" all day under load without stopping. If you are curious as to the specifics of what I do let me know.