On the other hand, Twitter also has a right to free speech. They're perfectly within their rights to promote their own opinions or viewpoints. I would assume that also includes the right to selectively amplify the voices of other based on agreeing with their viewpoints.

Because Twitter is a platform for the communication of others, they find themselves in an ironic situation: adhering to the principle of free speech means giving up, or at least severely restricting, their own ability to exercise that right. I think it would be noble of them to accept that restriction voluntarily, but it would be problematic if it were required of them.