Non-US-citizen here: Isn't the first amendment protecting private people and entities (e.g. newspapers) from government influence into free speech?

Private companies deciding not to service the head of the executive seems to be nearly the opposite of that, unless I understood this wrongly (which is entirely possible). Think of a newspaper (=private entity) decides to boycott a president and not write anything about him. Wouldn't it be the polar opposite of free speech if the president then had the power to force them to write about him?

I get that a payment service is not the same as a newspaper, but it is still a private entity.