Your phone carrier is a service provider that is constrained by the FCC. Twitter is one of many broadcast sites one could go via a phone carrier. But the US, to my knowledge, has never recognized anything approximating a right to broadcast. Not even freedom of the press implies freedom to spew signal over the airwaves, for instance (because the radio spectrum is finite and shared).
Freedom of the press has never implied freedom to use someone else's press, and telecommunications hardware providers aren't in the same category. If your phone carrier cuts you off, nobody on the Internet can read your signal; if Twitter cuts you off, they've merely taken their megaphone out of your hands.
We don't have jurisprudence saying so for websites, because this is new, but it could be argued that a number of websites have reached a sort of monopoly status. Facebook is probably closest, but Twitter and Wikipedia both occupy unique places in society to the point where it's not, just, "go start your own website".
As far as business freedom, remember the gay wedding cake guy? Where were all these "you're not entitled to Twitter's megaphone" people then? On the exact opposite side of the principle, mostly.
I think people having different attitudes regarding the three categories of service is reasonable.