Killing elk doesn't require a big caliber rifle, but it does require one you can shoot well. Quite a few elk have been taken with cartridges I wouldn't try, but if your patient and disciplined enough to take the right shots then just about any rifle will do. With current bullet construction a lot of different cartridges are now capable of providing better terminal performance than ever before.
You don't have to just use the .308 or .30-06, and like others have said if recoil is a problem use a lighter bullet in the 150-165 grain range instead of 180's. I'd probably go with the Winchester M70 as well if your budget can afford it, not that the Marlin isn't a good rifle they just aren't the same class so to speak. Winchester offers a .270 Win, 7mm-08, .308, and .30-06 all of which are very versatile and adequate for elk. I'm sure the Marlin is offered in the same chamberings as well.
For chasing elk in the Utah mountains I feel the M70 FWT or Extreme weather to be the best offerings for rifle choices. I really like the FWT's but the EW is really starting to grow on me. The M70 Super Shadow will be the cheapest Winchester around $600 new. Marlins will be in the $350 price range at most places I've seen them. Don't overlook good used Winchester push feed M70's and M670 rifles they will sevice you very well and be a lot cheaper to purchase.