While there are certainly a lot of great cartridges out there for hunting medium game, my personal preference is and has been for about 2 decades the 308 Win. I have and still do hunt with a myriad of others from time to time ... from 6.5x55 Se to 444 Marlin.
The 308 Win. coupled with a quality 165 gr. bullet, the round is bad medicine on deer out a lot farther I can proficiently hit inside a 10" circle. And I consider my self a decent shooter.
Where I hunt, from tree stands and blinds, here in the SE USA my typical shot seldom exceeds 150 yards, and more often than not is inside 100 yards. About any decent center fire cartridge will kill deer at these distances. But I usually tote my 308 Win.
No doubt about it, the most popular 30 cal. cartridge down this way is still the 30.06 Spr. I don;t see that changing anytime soon !!! I have no arguement with that at all. Great and very versitile cartridge. In fact a Browning BBR in 30.06 Spr. was for years my favorite hunting rifle.
As far as short vs. long action? About the only advange of the short action that I think comes into play is a tad less weight and shorter overall length because the receiver can be shorter.