Echoing the sentiments elsewhere in the thread, the office of the President has official avenues such as the White House Press Corps and EAS. I don't think Twitter has a particular duty to preserve or distribute the content of any user no matter who they are. It might be different if this was in the terms of service, or if it was something the executive branch was paying for.

If they changed their business practices to the model used by other network service providers commonly afforded neutral carrier status (i.e. SaaS, ISPs), then I think there might be a case for your second point.