My view depends on the scenario. Where I live there are rules about what guns may be used in certain places. For instance rimfires can't be carried in certain areas (not powerful enough, and they possibly want to help prevent the shooting of protected birds). So, with things as they are I would have a .22 and a larger center-fire rifle. The center-fire I use mostly nowadays uses the relatively abundant 7.62 x 39 cartridge. Not a wonderful performer compared to some, but perfectly adequate to shoot a deer for the freezer.
But if I had to survive with just one gun, I'd pick a robust .22 rimfire. By robust, I mean a sturdily built rifle with a simple, strong mechanism and securely fixed sights. If there were big animals to worry about, a repeater.... maybe even a semi-automatic.... would be best. The old steel-action Gevarm semi-auto seems to be a wonderfully reliable gun (until you get a dud cartridge). It fires from an open bolt position, and the 'firing pin' is a simple chisel edge on the front of the bolt/striker that makes a deep dent right across the diameter of the cartridge base. I've taken some big animals with a .22 rimfire, although I do not recommend it for big animals.
Head shots with a .22 can be very effective, but I'd rather call these 'brain shots'. Just shooting an animal in the head does not guarantee that it is going to drop. A shot to the jaw or anywhere other than the small brain area is not going to do the required job. I believe that even if your bullet hits the skull on the outside of the brain cavity, it can still be deflected by the bone and you can have an unpleasant situation. I have heard of bullets richocheting off a goat's forehead, and I have experienced some disasters myself. A while back I had some wild pigs to shoot on farm land where it was not wise to use a big rifle. I found that shots placed in behind the shoulder were the most reliable way to make sure that you got your hog. I just could not be sure that a head shot was going to reach the brain. More recently I killed and butchered some sheep on a farm. One dropped instantly with a head shot, the other didn't seem to know it had been hit .... there was no sign of distress or anything and it just continued wandering around in the paddock (although it was bleeding). I had to shoot it again. I didn't do an autopsy to see what had happened to the first bullet, but nevertheless it was a sobering lesson.
A shotgun can be adapted to hunt most things, but will you have the right cartridge on hand when you need it? Plus they make a heck of a bang and have a relatively limited range (which is a good thing where there are possibly other hunters nearby). The cartridges are huge and heavy. I have a shotgun, but I don't use it much.
I'm not a fan of combo guns, although some folks like them. If I am hunting game that requires a shotgun, that is what I'll take. If I need a rifle, then I will just carry that rifle and ammunition.
You could have a gun with interchangable barrels, but that is still more than one item. So why not have different barrels with the stock and action attached to each of them? If one of the guns develops a problem, you can use the other.
Still, you have to follow your own inner desires. Let us know what you decide. Best wishes.... Coote.