I would argue that it is not a *hard* 1st amendment issue. It is a "soft" one. A hard issue would be forbidding certain kinds of communication explicitly. This is a mere implicit one.
A corrolary would be to say, for example, Facebook wins the social media world hands-down. They own all social media platforms, and have wormed their way into controlling even legacy methods of such digital communication, like email, IRC, and pals. If you want to communicate text over the internet, you have to use Facebook. Now-- Throw into that an onerous TOS that forbids you speaking about certain things using their service.
Suddenly, you have massively curtailed ability to share expressions of ideas freely. Facebook is not a government agency, so "It's not first amendment!" chorus. But, is freedom of speech restricted all the same? Yes--- Is that power ultimately descended from the complicity of the government, via enforcement through the courts? YES.
Similar concept here. Disney would never show a political film detailing the very lengthly history of abuses the company has performed against artists, performers, and those critical of the brand, for instance. Such speech is a legally protected form of speech in fact-- but Disney, being a private company, has no obligation to show such films, and in fact, has every reason to prevent such knowledge from being disseminated. Giving full control over both production and distribution of a form of media to a single actor, is absolute nonsense, if your stated objective is to assure the free and open expression of ideas, which is what the 1st amendment codifies.
Specifically, the violation of the amendment, is the permissivity of the govt to permit this condition to manifest, when the 1st amendment asserts that the government is not allowed to take such an action. It is still a 1st amendment issue, just not a hard one. It's a soft one, that occurs because of permissivity of government for a 3rd party to perform the restriction instead of doing it itself. The power to prevent the outside messages being broadcast still ultimately comes from the govt, via the courts and penal system.