Many UK "private" courses are actually more open to the public than their US counterparts, correct me if I'm wrong.
British Open rota playable by public:
St Andrews, Carnoustie, Muirfield, Turnberry, Royal Troon, Royal St Georges, Royal Liverpool, Royal Lytham, Royal Birkdale
US Open Courses playable by public:
Pebble, Torrey, Bethpage, Pinehurst
US Open/Major courses not playable by public:
Winged Foot, Augusta, Olympic, Congressional, Oakmont, Olympia Fields, Merion, Baltusrol, Medinah, Hazeltine, Shinnecock and many more.
Bethpage and Torrey are only recent entries to the US Open as well. Fortunately, there are a wide variety of courses available to play publicly in the US so this isn't that noticeable.
But one thing I found about playing in the UK, if there's a course you want to play, chances are much better I can play it than in the US. UK is so much better in this regard.