Depends on the range you're going to shoot deer. If we're talking 300 yards and under, I think there's no appreciable difference between all of those in terminal ballstics with quality bullets.

What's to like about the .308?

- It's a very well known cartridge. If you're looking for load data, experiments for best groups, you name it - it has been done with the .308

- There's a ton of once-used brass available at very reasonable prices. I'm partial to Lake City 7.62 NATO brass or the European brass (Lapua or Norma). If you load reasonably (ie, not trying to load it super-hot), you can get at least 10 reloads out of .308 brass.

- there's a huge number of .30 caliber projectiles available for everything from target, plinking, large game, varmint in .30 caliber, because there's so many cartridges that launch .308 projectiles

- Every sort of rifle is chambered in .308 Winchester - bolt, falling block, lever-action, semi-auto (Remington 760 to the newer AR-10's), slide/pump (the Remington 740, aka "the Mennonite Machine Gun" as my gunsmithing school buddies from Wisconsin called it), full-auto.

- The .308, unlike some of the other cartridges you listed (in particular, the .257 Wby, the .270, and especially the .243) isn't "over-bored" - ie, they're not trying to shove a huge amount of powder down a small bore, which results in throats burning up. A .308 rifle barrel should last you at least 3,000 rounds. Some of the smaller bore, "hotter" cartridges are burned up in as little as 1,200 rounds (I've re-barreled .243's that were burned up pretty good in the throat in 1200 rounds).

- The .308 in a short-action rifle, gives you a stiffer rifle, and the potential for smaller groups over a .30-06.

For hunting there's absolutely nothing wrong with either a .308 or 30-06. They'll both do the job. With a careful choice of modern powders, you can close the velocity gap between the .308 and .30-06 to about only 200 fps. 

Even though I'm in the town where Weatherby is now located, I tend to avoid Weatherby cartridges. Their brass is expensive, harder to find, etc. True, they make some cartridges that are hot, hot, hot and push the limits of what you can do in a bolt gun. I'm old-fashioned when it comes to guns - I like my safety margins.

I second the suggestion to look at the 7mm08. It gives you the option of using 7mm bullets, of which there are as many good choices as in .30 caliber, it's not over-bored just yet, easy on the recoil, and in the same bullet weight (eg, 160 grains) you get an increase in sectional density and ballistic coefficient (meaning it shoots a tad flatter). If you were looking for the same thing based on a .30-06 case, you'd be looking at a .280 Remington.