Your distinction is correct as far as it goes, but there is a third observation that must also be taken into account, which is that the tidy division of economic, social, and political matters into private and public domains is uninteresting from the perspective of power. An example of this is how gov'ts, constrained by law, cannot legally engage in widespread surveillance, have found what is morally a loophole by involving private companies who are able to do so more freely. Or, for example, the policing of certain ideas on privately-owned social media which have become de facto public forums. It is completely uninteresting to claim that you aren't forced to use these forums or that you can start your own as jejune free market extremists like to claim. A little intellectual maturity will make plainly obvious why that is the case.

Now, I happen to think that, pace Dershowitz, adult content does not fall within the scope of free speech. The entire purpose of free speech is to allow the truth to be to expressed. Free speech takes an attitude of pragmatic permissiveness toward certain varieties of what are objectively bad speech as the price for that to happen. It's a choice that was made in American political history, but even here, the bounds of what is legally permitted under free speech have not remained fixed for various reasons.

Adult content is nowhere in the vicinity of this notion of free speech, and certainly not its moral purposes. There is no right to produce or to view adult content. There is no right to anything that is objectively unethical, and both the production and consumption of such content is unethical. Gov'ts can choose to take a permissive stance toward such activity for prudential reasons (for example, historically, while prostitution was categorically condemned on moral grounds, gov'ts took a permissive attitude in some respects, because they felt that banning it would cause still greater problems), but they have the authority to criminalize it.

So, given that it isn't a free speech issue, I have no problem, in a free speech context, with private companies banning such content from their platforms or from being the subject of transactions passing through their systems.