In Scotland at least, I don't know about the rest of the UK, the use of large amounts of land for sporting estates (grouse shooting, stalking deer) probably isn't so much of a class issue but more about it being part of a very very long history (~1000 years) of land ownership issues, clearances from land, access rights, opaque ownership structures....
Also these outdoor sports are hugely ecologically unsound - driven grouse shooting (shooters stay still and are driven towards them) requires large amount of land to be burned each year so that young plant shoots grow to feed the grouse and tracks bulldozed into the hills, deer stalking has increased deer numbers so much (combined with other things like natural carnivores long been hunted out of existence) that natural tree cover has been terribly reduced across a lot of the Highlands.
Having said that, things have got a lot better in the last couple of decades - with "Freedom to Roam" landowners have got a lot more sensible about allowing access and most support it with a lot running side businesses with accommodation, selling venison etc. Also there are some incredible projects like Trees for Life aiming to rewild the Highlands and the Scottish Government and other agencies are encouraging community ownership of land - which can only be good. I should also say that there are few very wealthy people helping out in these areas as well.
So it's slow progress but I feel that things are definitely going in the right direction!
>> natural carnivores long been hunted out of existence
That is a trend seen in island nations. Japan and the UK both wiped out the large predators. Some speculate that when a country owns and entire landmass that the one government/culture gains control sufficient to enforce its will absolutely. Don't like bears? They can be wiped out with the passing of one law. Where landmasses are divided, no one government/culture has power. Take the US/Canada landmass. Countless US animals (ie wolves) only survive in the US from refuge populations in Canada. Had the US culture owned the entire landmass, as in the UK, then wolves would likely have been eliminated.
The lasts bears and lynxes were killed in Britain a long time before there were anything like unified political structures - indeed possibly even before the Angles (English) and Scots even arrived on the island. The last wolf was killed much later - usually given as the 17th century but even that is before Scotland and England were politically unified.
These areas are not nature parks. They are modified/burned to support grouse. Natural predators are trapped and farm-raised animals introduced to be later shot. A hunting reserve is not a nature preserve.
