Oartly, it also depends on just WHERE you will be hunting. I would go for a .270 because you can load it with 130-grain slugs and, around here, you can often NEED that range, especially for shooting deer on the sidehills of the river valley, where a 500-yard shot can make or break your hunt. As well, the .270 HAS the reach to take an animal on the other side of the quarter, and do it with 1 round. You can't say that for the .243.
Sure, it's a long action and it's an old cartridge (came out in the 1920s) and it is based on the old .30-'03 casing (roughly 2mm longer than a '06). But if you want ONE rifle that will do five-eighths of ANYTHING, with the .270 you will have that versatility. You can load 130 for jumper, 140 for booboo, 150 for MOOse and you're in business. Put enough glass on it and load with 110 or 130 and you have a fairly decent gopher gun, too, although we generally use a .22RF on them, at least out to 125 or so yards. Then the big guns come out and the ranges increase.
The .270 is an old cartridge but it has the case capacity to be loaded for just about anything. Powder technology has improved a lot since it was introduced and the newer powders enable you to get the full potential of this fine old round without going to utterly-silly pressure levels and burning out your barrel in 500 rounds.
A very close friend of mine just gave away his prized 1948 FN Mauser in .270. He bought it, second-hand, in 1954. Last month, it took yet another moose. Original barrel and a LOT of deer between those two events.
Only other calibre I can think of that is almost as useful as an all-round gun is the .303...... and likely you don't want one of those, anyway.