https://papers.ssrn.com/sol3/papers.cfm?abstract_id=139661
The FCC has some guidance on free speech as pertains to broadcast (television, radio) media:
https://www.fcc.gov/consumers/guides/fcc-and-freedom-speech
"Individual radio and television station licensees are responsible for selecting all broadcast matter and for determining how their stations can best serve their communities. Broadcast licensees are responsible for choosing both the entertainment programming and the programming concerning local issues, news, public affairs, religion, sports and other subjects to be aired by the station."
The EFF's issue centre on free speech is conspicuously silent on the specific matter of major private-platform provider rights and/or obligations:
https://www.eff.org/issues/free-speech
The Net Neutrality debate touches on this question in part, and the pro-net-neutrality argument generally contradicts your viewpoint:
https://www.aclu.org/feature/what-net-neutrality
Much of the question revolves around the definition of "common carrier", and who's included. By one argument, a constitutionally protected service providing common access to billions of people, might just conceivably fit that definition.
https://www.nytimes.com/2014/05/11/business/defending-the-op...