Hey Red Label, Now I've got enough to work with and understand a bit better what you have in mind.
Actually, all the previous answers are "still good".
Here is the problem, hunting Big Game (like Elk) at a very long distance (over 500yds) is real difficult for the folks that have the correct equipment and spend many hours both on the Range and Afield learning about trajectory, wind and concentration. Now, the "best" rifle to accomplish this task (between the two you mentioned) would be the 300WinMag. Even better would be a 30-378Wby. However, the BIG problem is learning to properly control them.
Let me give you a short example, it would be like buying a 1200 Harley for an 8 year old that has never ridden a bicycle. Now, it could be done, but it is the absolute "hardest" way to go about achieving the goal.
Johnny mentioned shooting a buddies rifle to see what you are getting into, and that is an excellent idea. It would really be best to get a 308Win and wear out a barrel or two before you try to get where you are going. Once you do that, you will have enough "experience" judging the environmental conditions to "consider" Long Range shooting at Big Game.
But let me address some of your answers:
1) 500 yards+
2) deer/elk/bear/smaller game
You need to consider the toughest critter in the bunch and that would be the Elk, unless you meant Grizzly. And some folks do kill Elk beyond 500yds, but they have many, many years of "trigger-time" in the field. Here though, the "Best" answer concerning equipment to accomplish this task is a huge case of powder, big sleek bullets, a well built rifle, a very high power scope and a Lazer Range finder.
3) anywhere/everywhere
This means you will be faced with temperature extremes, rain, snow, fog and wind seemingly coming from every direction at once. Only shooting Afield will get this experience for you so you have any chance at all of making a clean kill.
4) neither
This goes back to the Harley/bicycle example. It really is best to work your way up. Otherwise, you will really disappoint yourself.
5) not yet
You need to begin Reloading for which ever rifle you decide on as quickly as possible. It will make you more "self-reliant" (which seems to be a concern for you) and will get the cost of ALL that huge amount of ammo you are going to be firing under control.
6) 2-3 times monthly
If by this you mean pistols/revolvers on a range, that is a good start, but not what you need for the long range shooting portion of your desire. You need L-o-n-g Distance shooting time with the rifles on the Range and Field time with the rifles just shooting at dirt clods, rocks or Varmints at various "unknown" distances. Time Afield is a must.
7) lots, but probably only several hundred max of this
This is good. The more the better, as in 1200-up. And as said before, Reloading is the best way to be able to afford the amount of rounds you will need to shoot.
8) an eastern woodchuck, or a... LOL
RL: I just want to aquire and have on hand a long range rifle that could serve me in many capacities should the need or desire ever arise.
HC: Interesting idea, but you need to start practicing as quickly as possible if you ever want to be able to make those 500+yd shots. If you are determined to the above, go with the 300WinMag, but I'd recommend the 308Win to start. By the time you get good with a 308Win, you might want a 338-378Wby or something that has not even been released yet.
RL: I don't hunt now. But would if I had to feed my family.
HC: This is a serious problem. Hunting like you are talking about takes years to learn. A few folks get beginners luck and go kill a Deer on the first outing and you will see those stories in some of the Gun Rags. But, Hunting is not something you do like going to the grocery story. It just takes time to learn the methods.
RL: I'd like to have something that could do the job cleanly and quickly. Not into causing animal suffering, or tracking wounded game. I want to get one while I still can...
HC: I completely agree with those concepts. I'm really not trying to discourage you. It is just that you are trying to reach the "end" before you have gotten to the "beginning". And there is a lot to learn along the way. The quicker you get started, the better.
Good hunting and clean 1-shot kills, Hot Core