The extent to which corporations should be considered private entities does not seem like a settled question to me. Your corporation has no natural right to exist, and therefore does not have the natural right to do business solely as it pleases.
> and you have about the same level of responsibility for your party as twitch has for their platform.
What responsibility are they exercising? If the behavior on their platform is not disruptive to it, then what precisely are they defending their users _from_?
> You probably also don't care what the justice system thinks about the invitations to your party.
Invite someone on the most wanted list. I can't imagine it playing out that well for you.