Earlier quoted context omitted.
I think it's even more concerning than that. Threatening to shut down private companies -- not for limiting speech, not for refusing to distribute speech -- but for exercising their own right to free speech alongside the free speech of others (in this case the president). There is no right to unchallenged or un-responded-to speech, regardless of how you interpret the right to free speech.
Attaching a disclaimer to the speech of another though is not straightforward. Will they get into the business of fact checking everyone over certain number of followers? Will they do it impartially world-wide? How can they even be impartial world wide given the different contradictory points of view, valid from both sides? Cyprus? What’s the take there?
But whether or not you like the feature, the idea that the president of the United States would threaten “shutting down” Twitter because he doesn’t like a feature is beyond the pale, full stop.