If that forum is censored with too broad a stroke, that actually harms society. I think everyone can agree that too broad censorship is bad, and if it hits large swathes of society, is really bad.
The difference between youtube and hacker-news is scale and near-monopoly. There is also an important difference of human and machine censorship. Not a difference of appeal, but a difference of unintended consequences. A human censor will notice when certain rules have unintended and too broad of an effect. There is a baseline of human oversight built in because there is a human somewhere. Automated systems do not have this property, if they have rules that seemed reasonable but aren't in practice, or aren't anymore, the system won't notice.
Like the other responder said. Youtube isn't just "someone's site". It is a full half of the bloody internet. The other half being split between Google and FB. I exaggerate for dramatic effect of course. But, most of the video content consumed online is currently on Youtube. And most of the internet is solely accessible through Google. Not to mention that a full 1/8th or 1/10th of the World has Facebook accounts. Funnily, I just Googled it, and according to the first hit Facebook has 2.2 Billion accounts.
At some point we have to admit/define/state that these entities and the products/platforms they provide are utilities for lack of a better word. Once they get to that point, they should rightly lose all ability to take sides. They must be hit with a giant stick if they so much as think of using their platforms to do anything other than make a profit. Imagine if telephone companies started dropping calls or disconnecting users that were saying "naughty" things.
Dan does a great job of moderation here, and if you break the rules repeatedly, you'll get kicked out. I guess that's "censorship" if you squint, but it's also YC protecting their private property and their rights to control who does what here. I don't see how that's a problem?