No, I understand just fine. You think that there is no chance of anyone ever walking into an area of a public park. My point is that no matter how much you say otherwise, there is always potential for someone to go there, even if it's not for the game. If it's an area of the park, it's possible for someone to go there, and therefore you should be cleaning up after the dog.
It's not about what your dog will or won't do, it's how other people will react to it. There are people who are so terrified of dogs that they will freak out at the mere sight of an unleashed dog. Should they not be allowed to golf and/or enjoy the park in peace?
Let's say you're playing a round by yourself one day, and the course is pretty empty. As you're coming up on the 10th tee, you see a big tiger sniffing around. How much would you freak out? Me, I'd be running off that course. Doesn't matter to me that the tiger is someone's well behaved pet, It's a friggin tiger, and I'm out!
Yeah, this seems like a ridiculous analogy, but it's similar to how some people feel about dogs. So why put them in that position?
I've had some extremely well behaved dogs, but I would never even consider taking them in public unleashed.