As a nerd who keeps data on all of his kills... I absolutely will recommend a .308.
There is no free lunch in anything. The .308 will generate more recoil, and be ~1.5lbs heavier to carry. I use a three chamber brake, so the recoil is really no different than the 6.8/Grendel guys who run suppressed...but the tradeoff is that I have to wear muffs and I hunt alone.
I can get more running pigs on the ground with 30 cent .308 bullets than I can with 80 cent 6.8/6.5 bullets. It just is what it is, and more related probably to energy dump than anything else. YMMV, but my data is conclusive to me and my sample size larger than most. Ironically, the .308 is cheaper for me to reload and hunt with.
If you subscribe to the ideas of hydrostatic and hydraulic shock, the .308 will give you more of what you're looking for over small frame AR cartridges too. If you think that those are hogwash, then disregard completely.
The 6.8 and Grendel guys argue amongst themselves a lot. Having killed 100+ pigs with each, I find their performance on game to be virtually identical. It will come down to bullet selection (because shot placement on runners is a wish) on which does better for you. I see zero reason to switch if you have one or the other. I have both. I like both. Both are extremely underrated.
I've killed 50+ with a 5.56. It does just as well as anything on opening shots where you can pick your placement. It does comparatively suck on runners though, as you will lose an additional 15% or so over the other cartridges I mentioned. I've put some monsters down with the 5.56 though.
Adding to the confusion are the heavily edited videos that fload around the internet. Watch the hunting videos where the editing has a lot of hard cuts between scenes when there are still pigs in the field, but you jump to the next scene...that is because either the guy missed 6 shots in a row, or he tagged the same hog poorly 6 times and it got away. If that guy is trying to promote something, realize that you are getting a heavily edited version. But it is the internet...nobody misses, and everyone does it better than you.
Oh, and I've found that killing 8-10 hogs with a particular combo can actually be pretty misleading. You might have a streak of fantastic luck. All of a sudden hogs 15-20 are made out of Kevlar and you're left scratching your head.