I don’t think accessibility makes a golf course better. It is unlikely many people played the original Lido, but that doesn’t mean it wasn’t great. Then too, St. Andrews has long been accessible, but it is the golf itself that is great not that you and I can play it.
As for Pine Valley, regardless of its private status, the place probably couldn’t handle much more play. It is too near a major city and the demand to play it is so high. Bob Kain of IMG, a member and one of the most powerful people in golf once told me he couldn’t get a tee time in June when he called in February!
My argument here probably gets into a much deeper philosophical point than most are interested in. What we mean by "greatest course" is, as I argue in the relevant comment, effectively nonsensical. However, if I'm meant entertain the question, my point is that insofar as we mean "greatest", how would we understand that.
There are a few ways to look at this. The way I see you, and many, talk about what a "great" course is, typically is context-free. That is, from a hypothetical third-person perspective, if one were to be able to play the course, in an idealized situation, the course would be the most pleasant to play (once or generally) of any other course in existence.
The problem with this perspective is that it must exist with this type of idealization built in. If you
can't, in actuality, play the course, or if when you play the course, you're subjected to some sort of harassment by the people at the course, it wouldn't be a good course.
To put it another way. Suppose one was debating the greatest city in the world, and not the greatest course. Now also suppose, for this thought-experiment, that person was homosexual. Could they honestly consider a city like Tehran as the "best" city, when from their perspective, living there would mean almost certain severe persecution? Or should they consider Tehran as a potential "best" city insofar as it were an idealized version of the city that it is, where if they lived there everyone would allow them to live the life they wanted to live, or perhaps they could imagine it as though they were different, and not the city.
Edit: I have thought of a better example of what I'm talking about. Imagine we are ranking the best house in the world. Let's say the house we were reviewing, was, say, an amazing mansion that had all the accouterments that we could want, however the house is at the South Pole, would we consider it the greatest house in the world? Obviously nobody would want to live there. However, if we are ranking houses, does the location of the house matter? It certainly matters for the real estate value, it certainly matters for it's desirability as a functional place. It's only when we remove the house from the context of the world we are in does the house have any value. We must imagine a fictitious version of the world in which the house isn't actually the house we are talking about, but is an identical house in another place. There are a lot of metaphysical issues here, but my entire argument will be that we choose to remove certain parts of the context of a thing, but not others, we remove disadvantages of one course without compensating the others. A mansion on the South Pole can have the location waved away, but a town-home in NYC cannot somehow have it's small size waved away. The problem with using an idealized version of a place, a course, is that we aren't really evaluating the course as it is, but as it could be. Now we must note that this idealization only works in one direction. While people regularly talk about what it would be like to play Cypress Point "as a member" in evaluating it, but hardly anyone talks about what it would be like to play Pebble Beach "if it were a private club" in evaluating it. We don't rate muni courses as though they were better maintained, we rate them as they are. Yet industry folks rate courses that, say, exclude women, as though they would be played by a women without harassment or exclusion. We only resort to an idealized view of a place when we have to. Though on it's face it seems as though a arbitrary third-person perspective seems like a sensible way of rating a place, we can see that it skews ratings of private places toward an idealization.
I think the real place where folks that present a context-free version of a rating system run into trouble is with regards to reproductions. If we truly have a detached third-person perspective, it should not matter whether we are playing the original course, or a reproduction, since the course we are playing is identical. A publicly-accessible copy of Pine Valley should be exactly the same greatness as the original. To argue any differently would be to start considering the context and history of a place, not
just the course, as authenticity shouldn't matter.
I think it's better to look at courses, not from some idealized version of the course, but as they are. We take the experiences that the course creates as the source of it's greatness. This idea of greatness is built on a much more utilitarian framework. It falls out that a place like Dornoch has a significant advantage over a place like Pine Valley in that it is able to create the greatest amount of joy for the greatest number. It also, awkwardly, falls out that physically accessible places have an advantage over inaccessible places for the same reason, and this is a fair criticism. The problem with this view is that it exactly is context-dependent, so the utilitarian framework is going to differ wildly from a personal perspective if that person's position is unique. A Pine Valley member will probably be up in arms at how their course is to being treated as though everyone who wanted had access to it, but it is exactly that the discontinuity of how much joy it could be creating vs how much it does that would pull the rating down from a utilitarian perspective. It's exactly the same reason that an accessible version would rank dramatically higher under this framework.
All of this is before we even consider the epistemological problems that are associated with considering a "greatest" course under a context-free version of a course. How much access does it takes to understand how good a course is? MacKenzie argued that even after significant play and close analysis of the Old Course, he was still learning from the long-time members different lines of play. This all leads to significant problems, but that's another wall of text for another day.