Of those listed, I would say the 338 will handle ANYTHING, the 30-06 is the most VERSATILE, but my 35 Whelen Improved will out perform the 30-06 in EVERY aspect. Even the standard 35 Whelen (not improved) out performs the 30-06 for trajecory, penetration/knockdown and retained down-range energy. The 338 Winchester is better yet, but pays the price in additional re-coil.
I am not very impressed with the 300's. I DO NOT consider them to be much of an impovement over a 30-06 with-in 200 yds. They tend to 'zip' through game without knocking them down. I have had Elk run a lot further with a 300 Ultra-mag than with a 30-06. However, for very open terrain the 300's would be a better choice.
In my opinion....I would rather have a magnum with a bigger bullet, NOT a FASTER ONE.
I find what Remington did with the Ultra-Mag very interesting (just a note) The 300 and the 375 UM's are the same length case. But the 338 Ultra-Mag is a shorter case, similar to comparing 300 Winchester, and a 338 Winchester. Now if Remington would just come out with a 350 Ultra-Mag with the shorter cartridge, I would be a happy camper. There is just something about the 35's, I think I am in love.