Eat it. Tan the hide. Sell it. Feed it to the dogs. Grind it for fertilizer. All are acceptable uses.
If you can get to them
I know ghillies, and I know hunters, and I know beaters, and they all complain at times about the long heft out with the carcase. Thing is that our country is very up and down, we don't really do much flat, and much of the up is rather inaccessible, and the deer make good use of it. Most of the older shoots used to take in garrons (hardy hill ponies) to help, but who has garrons nowadays ? so few folks keep horses at all.
They're not supposed to shoot them and just leave them on the hillsides, they're supposed to retrieve (partly to make sure the beast is decently dead) and everything is governed by 'sporting' laws, etc.,
Imagine dragging that carcase out by manpower....ask Neil or Mirage about just how much effort that is.
I know a landowner who was delighted to let an Italian shoot have at it in his estate for a week. He thought they'd redd out the deer that were cropping a new plantation. Nope. Instead they shot every wee bird in the place, from the robins to the blackbirds
Cultural differences abound.
While we still had wolves running free in the UK we still had much of the island forested. Good arable land was usually along river valleys, water meadows and the like. In came the Industrial Revolution and down came the forests,
Now we're trying to re-establish them where we can, but apart from folks who don't actually live on the land, most aren't terribly keen to see wolf packs back here.
We live on islands, busy islands, and predators are always going to be on a sticky wicket here.
No doubting it though, there are too many deer. I live in what is now suburbia, but we have a lot of woodlands around us still, and a lot of leafy paths, and we have a problem with deer. Never mind the greenfly munching your garden, the deer are happily working their way around, and the reality of traffic issues leaves horrified children and parents as something they see as cute is shattered into a gory mess. Two fawns and a doe last year at the end of the street, lot of tears and upset.
In a society that buys it's meat mostly from the supermarket now, deer aren't seen as food.
If put into packets and sold like beef though, it still doesn't sell very well I'm told. Folks think it'll taste weird, while we know the reality is that gamey is just old meat hung too long and that good fresh venison is just tasty meat (I have friends who supply my freezer for my husband)
How do they know the numbers ? I haven't a clue
I know that medieval records are not terribly reliable as an estimate outside of their own area. I know that driven hunts were often total overkill though, but the nobility weren't really much on restraint, just showing off, and rules claiming and enforcing hunting areas as no-go for everyone else.....especially after Rufus.