Free speech doesn't always mean an issue with the government. Free speech can be seen as the opposite of censorship.
This is plainly censorship (within Spotify's rights, but still censorship) and while nobody's 1A rights have been violated, it's still affecting someone's ability to publish to their audience.
Spotify already paid him $100M to make his podcast inaccessible to people. Where was the concern then? If Spotify limiting Joe Rogan from post the podcast to other services was not a concern then, why is Spotify limiting what it hosts now a concern? I see no possible philosophical distinction, but if you have potential justifications, please feel free to share.
I think that the fact that's he's been contracted for exclusivity really changes the nature of the free speech argument. Rogan is functionally being hired to Spotify to produce content. I don't expect the ability to freely express myself when representing my employer.
I think it's a very different situation, simply sharing your content on a platform, where I think free speech is much more relevant, vs. being contracted to produce content for a platform, where I think it stands to reason that you're expected to produce what the platform want.
Morals don't make money. Spotify has one responsibility and that's to it's shareholders. Companies that put morals over profits lose to those that don't.
This is plainly censorship (within Spotify's rights, but still censorship) and while nobody's 1A rights have been violated, it's still affecting someone's ability to publish to their audience.
That's not a legal issue, but a moral one.