Thlayli
Le Pétit Prince
Mostly I'm annoyed at how you've crafted your arguments. Use of inapposite analogies, appeals to non-existent legal authority, and similar fallacious arguments I find irksome.
As for "free" making the site somehow the property of users, I will note that YouTube (as one of many examples) is a "free" service with a community of users (frankly, it wouldn't exist but for contributions of content from the community), but in no way, shape or form does any member of that community have any of the rights you are asking for here. I'm happy to pile on with more examples, but I think the fallacy in that line of argument is clear.
In any event, happily enough, I'm not a decider on the site's PDMA policy. But I will say that asking to use the site's machinery to demand public explanations for moderator actions (or inactions -- in my view even more problematic, for reasons I posted about earlier) is unlike any other commercial enterprise's customer complaint resolution process. In the "real world," customers feeling aggrieved about some action (or inaction) by a company whose products they buy or whose services they consume can complain privately to whatever customer service, help line or ombudsman facilities that company makes available (here, it's the existing appeals process, depending on context). They are also (as you are) free to complain publicly and loudly, whether through social media or other outlets outside the company. What they cannot do is insist that the company's website be a forum to engage in a public debate about poor service.
I would say that what I have done is appeal to moral authority rather than legal authority. I don't think I'm filing any lawsuits any time soon, especially not with CFC so well-staffed with lawyers.

And in the end, this isn't a commercial enterprise like YouTube. It's a community forum built around the idea of community activities. My understanding (though you haven't clarified) is that the ad revenue is exclusively for supporting the site, and as such there's no attempt to make a profit here. Rather, the site exists for our benefit, for the benefit of the members, does it not?
If the site exists for the benefit of the members, and not for the benefit of Thunderfall or for you, the principles we're working from are substantially changed, aren't they? Do you work for Thunderfall? Perhaps in some sense. But do you work for us? Yes, you do. You're supposed to serve us, the CFC community. The reason you enforce these rules isn't because it brings Thunderfall a greater check each month, but because a forum overrun with chaos and spam isn't good for the community.
This is a good website. It's one of the best communities on the internet, which is why I have argued so tenaciously to make it better. And redefining everything in a legalistic sense sucks the life out of a community of people who are just doing what humans do: Trying to enjoy their lives.
The question is one of ideals, and of first principles, I believe. Our world's own democratic and liberalist theorists needed to begin somewhere, particularly in an environment of total autocracy where their own views were out of the mainstream. Whether or not something has legal justification, in the end, is entirely independent of whether or not it is right. New legal paradigms are created by wars, revolutions, and protest movements, that cause positions to change and entrenched authorities to relent.
Dred Scott was the law of the land with full legal justification, but that didn't make it right. Same for "separate but equal," and plenty of other outmoded and inhumane ways of seeing the world and individuals' rights in this world. But these legal concepts were abandoned when people began to choose to see the world in a new, better way. What matters, here and now, is that you and me, and all the community members, and all the moderators, have an exceptional opportunity to make this website a better place. We can make changes together that benefit all of us!
And again, this is not a company, is it? I'd like to see the registration documents if it is.