Regarding "has no obligation to host content it doesn't deem suitable": Yes, sadly, this is a problem not unique to YouTube. All hosters get to decide what to host and to change that decision at any time.
Some people choose to respond to hosting on sufferance by becoming sycophants (placate the hoster with speech one believes won't offend the censors) but that is ultimately detrimental to one's own message and free speech overall. We have no solution for this that is really easy for nontechnical computer users because we all depend on other parties for Internet access. If homes had Internet access that could not be taken away, we might be able to recommend self-hosting. For now, in the age of the ISPs, the best we can do is massively decentralized hosting which uses no DRM, running additional nonfree code (such as YouTube's DRM in JS), and somehow getting a lot of peers with lots of storage and bandwidth to keep available copies of everything uploaded (we ought not decide that one message is worth sharing and not another message because that is the very problem we're trying to solve).
This is a tough problem to solve. I don't know what a viable solution will look like quite yet. But I know what to avoid: we won't solve this problem by placating big business desires for more control over what we say and how we get copies of other people's works.