Weight and fit of the rifle are very important factors. And, some of the modern recoil pads are 25% to 40% more effective than those of even just 20 years ago.
Loading a .308 down to low recoil characteristics would (to me) mean velocities down around 2,000 ft/sec. This would still be adequate to kill Bambi at closer ranges. Say, around 100 yards.
For instance, I've long used 20 grains of 2400 in my '06, with the 169-grain lead gas-check bullet. This gives a muzzle velocity around 1,900 ft/sec.
(I've also used this powder charge in .30-30; .30-40, and .308. with 150-grain jacketed bullets. Works, although it's a bit more than "mid-range" in the .30-30.)
PROBABLY a mid-range load in .308 from an eight+-pound rifle, one which fits properly and has a very good recoil pad, would not be more physically disturbing than a .223 in a 6-1/2- to 7-pound rifle. MAYBE.
I don't want to get too optimistic about somebody else's eyes.
FWIW,
Art