Comparing apples to apples (especially factory loads) I'd take a .243 or even .223 over a .308 for 'accuracy' out of an affordable factory hunting rifle like a Howa with off the shelf ammo.
Two aspects to this, first the human one.
In the prone position where recoil is most noticeable I would not expect someone to shoot 100 rounds of 150 grain bullets in a session out of a .308 sporting rifle for practise, heck even 10 might be quite an ask for some.
So, it also comes down to how accurately the person can shoot the rifle. So why punish yourself with more recoil?
Bringing F-class rifles into the discussion isn't really relevant for stalking (not sure if you mean F-TR or F-Open? I doubt the .308 would be first choice in the latter category).
But let's talk performance.
Apart from a possible wild boar, I can't see any disadvantages to using a .243 or 7mm/08 (which is used on elk for goodness sake).
I had a 308 and sold it because I could not see any advantage offered by the 30 cal bullets when compared to their smaller, faster and sleeker brethren.
Without going into mindless BC babble, the smaller bullets will leave the muzzle faster and hold on to a greater percentage of that energy downrange. Sure, at 'UK hunting distances' of 300 meters or less the deer won't know the difference, but why not pick the right tool for the job?
The .308 is a bit of a blunt instrument IMHO... whilst the OP asked about 'like for like' shot placement, I'd still pick the scalpel over the sledgehammer (and what sort of surgeon only has one scalpel?)