There are a lot of good choices. I've killed some of all three with a 30-06, plus some of each with 6mm, 6.5x55, 30-30, bow, and probably another caliber or two if I think about it a few minutes.
Much as I love the .280, and would be tempted, and much as I'd love to twist up a 338-06 (which probably is an ideal for these three along with 35 Whelen and 338 Win. magnum), I'd probably stick with the plain vanilla 30-06 with one good 180 grain bullet and shoot everything with it. For me, that has been the 180 Swift A-Frame for elk and moose size critters.
I don't like the kick of magnums, and I do like big exit holes for blood trails, which .30 and up produce more reliably, especially with bonded or X type bullets. Elk are worth the better bullets. Less need for such bullets on the other two.
It's the combo that leans me toward a bigger hole in the barrel. 85% of the time you could do it all with a .260. Why travel on the edge when you can give yourself much more margin all of the time?
I just talked myself into the 338-06. YMMV