I feel the issue is not one of energy (appropriately placed; a firmly planted knitting needle has enough energy to kill a deer) but of accuracy.
Is the 308 suitable for hunting at 1000yds?
....well here's another way of looking at it:
If you are capable of making your wind call over the whole 1000yds to within 1mph; then yes.
If you are able to notice 1mph wind changes over the whole 1000yds ; then yes.
Otherwise; No.
Why?
(Taking Factory 175gr GMM as an example)
A 10mph wind will push the bullet 98.7 inches at 1000yds.
This means every 1mph change pushes the bullet 9.87inches at 1000yds (lets call it 10 inches for ease).
So
Even if you are capable of grouping with a 308 to 0.0001MOA at 1000 yds (and you won't be),
for every 1 mph you misjudge the wind, you will miss your POA by 10 inches.
Miss a 2mph wind change and your bullet goes 20inches wide of the aiming point.
...and that's forgetting the natural dispersion of your group compounding the error.
So, to answer the original question again :
If you're capable of consistently shooting your 308 within 0.5 MOA of your POA (ie shooting a sub 1 MOA group centred on your POA) at 1000yds and can make your wind calls and note changes over the whole 1000yds to within 1mph; then 'maybe', you'd be able to contemplate attempting a deer at that range, humanely, with a 308.
But it would be a poor and unforgiving choice (and the animals would the ones paying the price).
If you're not at that standard (and I'll wager there's not a man on this site that is); please forget using a 308 on game animals at 1000yds