I don't think the 300 Win Mag has much recoil at all. I have one and it is a blast to shoot. I took my Caribou in Alaska with it at 300 yards. In my opinion the 300 Mag has the advantage because of the larger bullets.
Nothing wrong with the 7mm Mag, but the 300 Win Mag gets the nod.
It depends on the rifle design. My M77 MkII 7mm mag really packs a wallop, while my brother's Savage 116 in .338 win mag kicks much less. Go figure.
I'd stick with the 7mm. I can speak from experience that it is enough for elk at the ranges you describe, will kill Black bears reliably at that range as well, and is WAY overkill on whitetails. That has been my personal experience. Considering the damage one 7mm mag bullet did on my last hunting trip, I don't want to think what a 150-165 grainer in a .300 mag would have done. :eek:
If you get the 7mm, as noted before, you ought to consider sticking with the 160-175 grain bullets. The 175s are great on elk, but may not expand reliably in a deer or smaller bear, while the 140 grainers do too much damage on expansion in a whitetail, IME. The 160 is a good compromise.
Like previous posters noted, there are better choices for Elk, but if I were to bump up from the 7 for Elk, I would go up to a .338 and skip all of the .300 magnums. Then again, my next "do all" rifle will probably be a 30-06, as it also meets the criteria you noted admirably. I do understand the novelty of a new rifle though, so whatever you buy, enjoy.