Only those cartridges?? Really odd! If they thought they were limiting this to classic cartridges, the .270's absence is striking. Maybe it's ok if you are buying your first rifle, but what if you own one already. I hunted the first 10 years of my life with an 8mm Mauser my grandfather gave me. He also had a .30 Remington, but none of the rifles on the list. While I now own a 30-06, my preference would be my 7-08 -- why in the world I could use one but not the other (particularly since the 7-08 is a necked down .308, which is on the list) is beyond me.
That said, if I was buying one of the approved ones for Whitetails, I'd probably go with the .308.