To be precise, actions by media owners that constrain or oppose *paid* speech (including false ads) are *not* violations of the First Amendment. Such constraints are well within the rights of all media outlets to deny unless they are 'common carriers' like telephone (or internet) providers.
Unless they are broadcast over public airwaves (which happens rarely anymore), any private- or corporate- owned medium has the right to dis/approve any paid promotional content from any customer for any reason. They cannot be forced to publish any 'free' speech (except perhaps Public Service Announcements coming from the government). However because denying political ads is likely to annoy many consumers, the airing of mainstream political ads largely unavoidable, even on subscription-only media.
The First Amendment ensures only that no non-commercial agent can *suppress* speech in a public forum -- unpaid speech in an unpaid medium. The law does not apply to any commercial (for profit) medium. Historically, because television and radio were broadcast using public airwaves, the FCC required they 'serve the public good' by guaranteeing *some* public access (like a community channel or late night shows where the public could speak out individually). But with the fading of over-the-air broadcast media, those channels of public free speech have largely been closed. At the same time, corporate media owners have steadily diminished public access to their outlets, such that I'm aware of *no* open forums it TV or radio that remain.
While the net has subsumed virtually all free speech that remains, any website may deny almost anyone access using means that circumvent lawful speech guarantee, such as violations of community guidelines, for example.
Now that all major media in the US is corporate- owned and public access to major media has been eradicated, in practice, free speech no longer exists in America. All that remains is net noise.