Well there really isn't anyone here who will make up your mind for you. We can all throw out what we prefer, and why, but that really isn't going to matter when your the one spending the money.
As for the criteria you mention, every one of the already mentioned calibers will do all of that. Now your not going to find a ton of match grade bullets in .277" as mentioned, but I didn't see where you mentioned setting up for formal competition anyway. With some work and tweaking most rifles can be made to shoot nice groups, and with proper bullet selection most can be used on just about anything in the woods of the lower 48 within reason.
I might seem a bit biased having four '06's in the safe, but they are in the company of a couple .243's, a couple of 25-06's, a 6.5x55, .270, .280, a couple of 7 mags, a .308, and a few others. I have hunted and cleanly taken whitetail, and mule deer, a cow elk, plenty of feral hogs and a whole bunch of coyotes with the mix and never felt under gunned. I have used all of those mentioned above to reach out to 400yds on deer or hogs and everything went into the freezer shortly after. Were they the absolute right tool for the job, probably not, but they all shoot nice 1" or less groups at one and two hundred yards, and I have always felt confident when I made the shot it was going to hit where I wanted it to. To be honest the one I haul out with me the most is my .308. Not because it is more accurate, or hit harder than any of the others, but because it only weighs 6.5 pounds, and is only about 36" long tip to tip. It doesn't sport a match grade barrel or a fancy delicate trigger, but it works and works well even out to the 350+ yards you mentioned, using standard off the shelf Remington 150gr Core Lokt ammo. With the short 16.5" barrel it is a far cry from what most would feel comfortable with, but it slides through the thickest woods cradled in the crook of my arm. It never snags and being so compact and being topped with the 1.5 x 6 power scope, it makes getting multiple shots on hogs running through the brush almost effortless.
When I look at picking up something new, I always price out the components first as well as look to see what available weights the bullets might come in and what the manuals have for listed loads. I very rarely if ever purchase factory ammo except when I purchase a new to me caliber, and those are simply to use as reference to what my loads should be doing. Another thing I do is look to see if I already have brass which can be used for the new rifle as well. As in the case of the .308, I can in a pinch use it for my .243's. Also if I finally decide to purchase a .260, I can use it for those as well. If I want to sit and over look my back pasture though, I like having the 7mm STW atop my Harris bi-pod. Hard to argue with sending a 150gr or higher weight bullet out at 3250fps and grouping around 3/4" at 200yds. For 200 - 500yds it simply rocks, but up close and personal it is a bit much.
With the '06 brass, in a pinch I can use it for not only my .25-06, but also the .270, and .280. Granted it might not be the best choice, but it is plenty easy to come by as range pick up, and it doesn't take much effort to convert it.
If you don't handload, and your buying factory, I would look to see what is mostly available in your area or that you can afford to order. In recent years the cost per box of premium ammo has gone through the roof, and there isn't any way I could see myself buying 20 rounds for more than $25 of anything. Especially when I can load those same 20, for around half or less, and kill my deer, hogs, or anything else just as quick and dead, just as far as with the high dollar stuff.
My suggestion is to go out and find a rifle you like, one that fits you well, has the right curves, looks and feel, then worry about the caliber.
Picking a firearm out is like picking out a date for the prom, there are a ton of good choices, but everyone has their own idea of which one is right for them.
