that's the point. America has green spaces, UK has country side, terroir if you will.
A green space is a delineated, commoditized destination. Drive there, park there, do your thing, go back. A country-side can be enjoyed through osmosis, even when living in the city.
They're qualitatively different.
Opinions on property rights aside there is no lack of land to explore and enjoy.
Aside from there there are also state parks and forests, though the states define their own terms of use and enjoyment around them.
It's less about delineated activities (which are a number, countable, very modernist), and more the day2day enjoyment of nature, as it bleeds through city life, you don't necessarily seek it out, it just happens (which can't be put into a number, can only be waxed rhapsodically about, very humanist). It's visiting family one town over, leaving your city house and smelling cow shit as you bike there. It's a train commute and seeing the fog roll over centuries year old pastures on the way. It's eating venison in a tavern restaurant in fall, and was shot by the local hunter's club. It's a date that starts as a forest hike in the afternoon and imperceptibly blends into a pub crawl at night.
I live in a city of a state which is by all accounts rugged and rustic, in close proximity to wilderness, parks, farmland, etc... but there's a clear separation of intent. When I lived in Europe, the enjoyment of nature was more through a surrounding vapor, unconscious.
I'm not saying that one is necessarily worse, but to me the experience is starkly different, and that difference can't be captured in numbers.
It's also a spectrum, location dependent, ymmv, blah blah etc..
There is something to be said for the UK traipsing laws from time immemorial - because you can explore something other than a million square kilometers of mostly unused land.
To me there has to be the people to give the nature significance. Otherwise it's not so much exploration as it is just a massive camping trip.