Interesting question, but far too open-ended to make a hard science out of.
Best hunting cartridge….for hunting what, and where? A 30-30 is outstanding for deer in Vermont but lacks the range many hunters would like to have hunting deer in Nevada.

I think it is arguable that the “best hunting cartridge” for most hunting in the USA is either the 22 LR or a shot shell. This from the fact that far more small game is shot every year than big game.
I know men in Alabama and Oklahoma who kill about 6-8 deer a year each, but they kill 10-15 rabbits and birds every time they go out hunting for them, and they hunt for small game 6 months out of the year.
So if I were to make a wild guess, I’d say the average American hunter probably kills 50-100 animals a year and some far more. If we look at the number of kills of all game compared to the numbers of kills of big game, I would bet the big game kills are about 2% to 5% of the kills.

In my opinion if we were to look at only big game cartridges most Americans (most being those that hunt white-tails and maybe pigs) the best cartridges for that task are probably the 260 Rem, and the 6.5 Swede. My reasons are these:

These 2 shells have all the power and range that could be asked for from a deer rifle. They also fit in autos, bolt actions, pumps, lever actions and single-shot rifles.
In the case of the 260, the brass is easily made from surplus machine gun brass, so it’s cheap. Too bad the old cheap surplus rifle powders are a thing of the past.
Loaded with 120 grain bullets these 2 shells kick so little more than a 243 that the difference is hard to tell. But they can be loaded with bullets of up to 160 grains too, so that pushes their utility far out in front of the 243.

I do not own either one.
Why?
I live in Wyoming and the “common game” here is the elk. Sure we have all the antelope and deer you could want too, and sure…the 6.5s will do for elk, but there are better choices for this area. That is not to say I’d be unhappy with a good 6.5, but having grown up out here, I have always gone for the 270 and 30-06 as general purpose shells.

I have used a lot of different guns and cartridges in my 50 years of hunting in many states and a few other countries, but I have not seen many offerings that show me much more than my original picks for how well they put game on the ground.

In my life I have killed big game with the following:
222, 223, 6MM Rem, 257 Roberts, 25-06, 6.5 Grendal, 6.5 Swede, 6.8 SPC 270 Winchester, 270 WSM, 7X57, 7X64, 30-30 30-40 Krag, 300 Savage, 308, 30-06, 300 H&H, 308 Norma, 303 Brit, 8X57, 8X68, 338-06, 338 Win Mag, 348 Win, 9.3X74R, 375H&H, 416 Taylor, 404 Jeffery, 458 Win, 50-140 Sharps, 45-70, 45 cal muzzleloader, 50 cal muzzleloader, 54 cal muzzleloader, 58 cal muzzleloader, and 62 cal muzzleloader , and in handguns I have made kills with 357 mag, 41 mag, 44 mag, 45auto, 45 colt and 454 Casull, as well as arrows from recurve and long bows. This is a pretty complete list, but I may have forgotten a few at the time I write this. In addition to this list I have seen game killed with probably 20-30 more cartridges used by friends and clients.
My point is simply that I am speaking from experience, not reading about someone else’s opinion in a magazine somewhere.

But coming back to the 1st question, we have to acknowledge that by far, most American hunters are deer hunters and will not hunt elk, moose, caribou, bear or buffalo, and if they do, most will not hunt them many times in their lives. So it would seem sensible to buy a special rifle for the special game, and use a general-purpose rifle for everything else. So my vote goes to the 6.5 Swede in bolt actions and single shots, and the 260 for lever actions and pump (if Remington would do it anyway.)

Now for the issue of “one size fits all” which is really more a dream then reality, I still believe the best all-around cartridge ever made for the hunter of north American is the 30-06.

If we go to the international idea of an all-around rifle, the title would go to the 375H&H.

The reason this is actually a pipe dream is simple.

Anyone that can afford to hunt big game all over the 50 states and also Canada as well as those that can go to Africa and Australia and New Zealand can easily afford more than one gun. So this scenario is interesting only because it makes us do honest comparisons of guns and cartridges as well as bullet performance, but in the real world we have to admit it doesn’t matter all that much, and all game hunting is covered very very well by MANY different cartridges.

Shoot what you like, use a good bullet that will not go all to pieces on impact, and you have a good rifle if you are a good hunter.

Use what you like and use it within YOU’RE limitations and you’ll be fine. Don’t shoot if you are pushing your luck. Just be responsible and have the respect you should have for the game.