Stephen,
I agree with Ian Andrew and Tom Doak: if you really want to visit and see the great (and private ) courses in America, you can almost always do. Sincerity and common courtesy goes a long way as Ian and Tom suggest.
A measure of sincerity is whether you insist on playing or, by contrast, make clear you want to visit to study the architecture and don't care about playing.
The first time I ever visited Crystal Downs was perhaps my most memorable trip to see a classic.
I just drove in and went to the first tee, a great place to take the whole place in. Very quickly a member came up and asked if I wanted to play. I politely declined telling him I was really there to study the course but would appreciate walking along with him if he didn't mind.
No problem, he said just wait until his wife joined him.
Now, to be honest his wife wasn't quite so welcoming. Indeed, she basically suspected I was trespassing and when she asked "what are you doing here?" I knew I was busted and knew I needed a good answer beyond just saying I wanted to see the course.
So, I told her the truth "I am on my way to Shinnecock Hills and just wanted to stop in" - Shinnecock being the final destination in my cross country journey.
Well, this lady knew damn well where Shinnecock was and insisted on asking "where was I coming from?".
So, again. I toldher the truth: "Los Angeles".
That was enough for her husband who wound up not just be very gracious on the golf course, but even invited me for dinner.
Not enough for the woman. She had at least two more questions:
Where was I from?
Where did I go to college?
So, again, the truth. : Greenwich, CT and Princeton. My interrogation was over!
But, from there it was a great experience discussing with the member each hole and how he played them. No pressure to play well. Just a very pleasant time to study the course and see why it was considered a hidden gem (as it was back in the early/mid 90s).
One final point: the comparison of golf courses to works of art only goes so far. One can spend all day at a leading museum and see lots of art. But, to study golf architecture you really have to get around. For instance, the two courses I haven't seen I most want to see are Hirono in Japan and New South Wales near Syndey.