Assume enough people want the government to regulate certain private online communities like Twitter, so that those spaces can be used for 1st amendment purposes.
This in my opinion is the strongest justification. This is just democracy at play, a large amount of the populace thinks this is important. If you find a way for everyone to agree on regulation, governments can put in place such regulation. It doesn't even have to be about any constitutional rights.
I don't think everyone agrees completely, me as well, I don't want to see these spaces become complete shit-show, full of trolls, bots, duplicate accounts, spam, foreign farm content, etc.
This is where I suggest a compromise, I think what I put forward is a good middle of the road, I think it can please both sides, those who want to be free to say whatever they want can, as long as they do so non-anonymously, and after having been fully verified to be a real legal person, have only a single account, and it's not illegal what they are saying. There might need some rules about the speech needing to be in context and not spammy as well, but that can be ironed out later.
Similarly, this should also limit the platform from the abuse that could come from just arbitrarily allowing everything and force laissez-faire attitude which would come from just say banning all form of moderation.
Now the question is, is there a constitutional issue that would prevent such regulation?
So we're flipping the argument now.
Nothing I can see in the constitution really protects the right of a private platform to not have the government make laws that can regulate how it manages its platform.
So being a private space doesn't really protect them from the people demanding regulation from government and for the government to enforce it on the platform.
That means, if we want, we could regulate these spaces so they can be used as an avenue for 1st amendment like petitioning to the government or redress of grievances, without interference from the platform private owning entity.
It also means, it's probably not currently illegal for these platforms to do so, since like you said, the constitution doesn't really say that everyone has the right to say anything they want about anything to anyone and anywhere. But also, this isn't really an issue, to me, this case isn't about what the constitution says or is interpreted, it's just about what we as people want for ourselves as laws/policies/regulations.
I think this could also be welcomed by the platforms. Currently, the lack of clear rules around moderation often put them in precarious PR scenarios, some amount of agreed on regulation can provide the political stability around their own line of business that could allow them to thrive even more.