Mind you, wars and sheep did have a pretty devastating effect on the Caledonian woodland cover of the highlands. The current population of the red deer aren't helping with natural regeneration. This is one of the reasons for the case for re-introducing predator species.
But that's a complex topic with no simple answers and easy divisions.
As you have pointed out the solution is to eat more venison, but most local butchers stock hardly any of it because of low demand.
British eating habits have become really narrow over the years. Its hard to find offal (healthier and greener than just eating muscle meat). Rabbit seems to have pretty much disappeared too.
Re offal I actually just did a really nice mutton liver curry the other day, something you won't find in your average "brown spicy glop for white people" takeaway but which you can get in the south side of Glasgow. Absolutely brilliant stuff, even better the second day if you can leave it alone that long. Four quid for about half a kilo of lamb liver out of the reduced section in the supermarket!