If we are defining "big bore" as over .30 caliber, I shoot and like rifles chambered in .35 Whelen, 9.3x62, 9.3x74R, .375H&H, and .450-400 3" Nitro Express.
The most versatile of them all for North American game is the .35 Whelen. but very limited in rifle and factory ammo choices. I have taken many Canadian elk and moose with mine with complete satisfaction.
The most versatile internationally is the .375 H&H. Rifles and ammo in .375 H&H are available anywhere worldwide. Big enough for the biggest game, small enough to shoot comfortably with some training and conditioning. I've shot everything ranging from Impala to moose and elk to Cape buffalo with the .375 H&H. Ranges from 300 M to rock throwing distance - it does it all and does it well.
The cartridge that is capable for shooting anything on earth, with mild recoil, and is available in a multitude of handy light bolt action rifles and with a wide variety of good factory loads is the 9.3x62. It can do it all, but is not legal for dangerous game in some African countries. It's my favourite. i've taken everything from small bushbuck to gigantic giraffe with mine. It doesn't kick much for such a capable cartridge and kills just as well as a .375 H&H. The 9.3x74R is the same, just chambered in double rifles if that's your preference.
The .450-400 3" or ".400 Jefferey" is my favourite double rifle cartridge. It is low pressure and has a proper rim for assured extraction. Once even more popular than the .375 H&H for dangerous game, it is a true giant killer, doesn't kick too bad, penetrates better than most, and is well respected. I've use mine for elk, moose, and a couple of Cape buffalo.
My experience with the .416 Remington is limited to one safari with a rented rifle. It was effectively employed to kill small Impala and medium sized wildebeest at ranges of 150 M or so, and stopped a full-on charge from a big bull elephant at 5 meters. Based on that one experience, if I was to choose a dedicated big bore stopping rifle with some extra versatile medium range capabilities, it would be a bolt action chambered in .416 Remington.