I have no data, but I would imagine very little of the land being discussed is owned by individuals who have generated sufficient personal wealth through their working lives, after paying tax, to purchase it. It's generally land they control through the pure luck of being born to parents with an estate, who themselves have, in all likelihood inherited it themselves.
There was one such woman interviewed on the Today programme on R4 earlier in the week. Owner of X0,000 acres in Somerset(?). It apparently keeps her awake at night worrying about members of the public entering her land and potentially being injured or killed by falling branches in her woodland. Not the threat of being sued, the actual injuries to people. I have never heard such BS in my life. I doubt she has any such concerns about injuries to her gamekeepers, beaters, horses, foxes or game birds.
Those of us who were lucky enough to be born or now live in the UK are, of course, privileged to have the access to the land we do. It's not the same in many countries. I see no harm in improving access to beautiful places in a respectful way. If I had a river at the bottom of my garden, I'd welcome people to walk a path along it. As it happens, we have one in a culvert under our drive. Cavers welcome, no permit required, no need to call at house. Depth 1m, Length : 70m. Grade 2. Avoid in wet weather.