frankly, if you go grab a remington catalog, and open it to the ammo page, and pretty much take anything from a .243 on up to a .35 whelen you will be well served. (Of course, the .30 carbine and the 25-20 and 32-20 are a bit weak, and should not be included, but you get the idea.)
Lots of elk are shot with .243's and .257 roberts and other "small" cartridges, and lots of deer are shot with 300 winmags and .338's as well,
What is the biggest dividing point on these uses is the design of the bullet used.
you can have three bullets, of the same weight, which will all perform as designed, which will be completely different in game. Lets take a
6mm bullet in 100 grains. you can have a Ballistic tip, which is almost a varmint bullet in the .243, opens really fast, shallow penetration and sometimes even in Coyote, will not leave the body leaving only one small entrance wound to sew up for the fur. Next you can have a 100 grain, VLD target bullet, with the performance of a near FMJ, drilling a perfect little hole in a deer, and as often as not, a perfect little hole on the way out. leaving a wounded deer with a good chance of running away far enough to die without being found. Finally, you can have a CT or Partition bullet in the same 6 mm 100 grain size, that will kill an elk efficiently if you do your job, and will anchor deer with perfection. All the bullets are the same "size" but completely different in use.
SO when looking at rifles, try hard to compare what loads are available and what those loads allow you to do. If you reload, then you can understand and use the flexibility of some rounds much better.