... there is zero chance that the president will be able to censor the private sector.
If you mean the U.S. private sector, you're right. But otherwise, the U.S. President is allowed to take a whole lot of actions which can effectively censor non-citizens, and I've got some bad news pertaining to one in particular involving compliance with European privacy regulations which could potentially result in the deletion of records including accounts of European citizens from hosting providers such as Google, Apple, Facebook, Twitter, etc. Please see:
https://www.engadget.com/2017/01/26/trump-signs-executive-order-stripping-no...
"Enforcing privacy policies that specifically 'exclude persons who are not United States citizens or lawful permanent residents,' while aimed at enhancing domestic immigration laws, effectively invalidates America's part of the Privacy Shield agreement, opens the current administration up to sanctions by the EU and could lead our allies across the Atlantic to suspend the agreement outright."
If Google is forced to delete all the personally identifying information of European citizens because the President ordered U.S. federal agencies to stop enforcing privacy policies, that would effectively be an act of censorship on a scale without historical precedent, would it not?