I have yet to weigh a short and long action of the same make/design and find more than a 4-ounce difference. This includes NULA's, Remington 700's, Savages, and post-'64 Winchester Model 70's. This is 1/4 pound, which is often less than wood stocks vary in weight. The reason some (not all) .308's weigh considerably less than .30-06 is due to other factors than action length.
If you compare different rifles, yeah, some .308's kick more than some .30-06's. In the same model of rifle the .308 will recoil noticeably lighter, despite weighing 1/4 pound less.
If somebody wants to use 200-220 grain bullets they're probably better off with the .30-06, but there's very little reason to for most big game hunting. Have seen plenty of big game animals up to "elk size" killed with the .308, and with today's 150-180 grain bullets there's no discernible difference. This is partly because the .308 beats the original muzzle velocities of the .30-06--velocities that resulted in the .30-06's great reputation on big game, back when iron sights were the norm.
At one point a number of years ago I owned six .30-06's, but finally realized I mostly hunted with one, the NULA I've used on more big game than any of my other rifles over the past 20 years. It's probably the most accurate .30-06 sporter I've ever shot, but have owned several .308's that were more accurate--and cost far less. Based on considerable time with Melvin Forbes, I'd guess he'd agree that on average .308's shoot more accurately than his .30-06's.
Part of accuracy, however, is how easy it is to get certain cartridges to shoot well. This is a controversial subject, because some shooters don't believe in "inherent accuracy." Others do. I've written more than one article about this subject, but David Tubb says the .308 is among the 3-4 cartridges easiest to get to shoot very accurately.
Right now my only .30-06 is the NULA. It's not only accurate but a great travel rifle, partly because ammunition that works well on big game can be bought anywhere ammo can be purchased. Due to the superb synthetic stock, and titanium nitride metal coating, it's also weather-proof.
But I also own several .308's, and hunt with them more and more, because they do the same things on 99% of the world's big game without kicking as hard. Despite my semi-advanced age I'm still not particularly recoil sensitive, but don't see any reason to get kicked harder if there's no reason.
The heaviest of my .308's is a pre-'64 Model 70 Winchester Featherweight weighing 7-1/2 pounds with scope, but my favorite is my Merkel single-shot, which despite its fancy walnut, weighs exactly the same as my NULA .30-06. In neither the Model 70 or Merkel rifle is "action length" a factor. Instead, they just shoot well and kick less than the NULA. I won't ever sell the NULA, and will still pick it for a lot of hunting, but the instances it's chambering would work noticeably better than any of the .308's would be very rare indeed.
