These sites need to enforce decency like any other media company
Which returns to an earlier question of mine: where's that line? What's the measure to draw it with? You said earlier:
We don't need to have a public discussion about what's permissible on private forums
Suppose there's a corporate coup tomorrow on, let's say, Reddit, and there are people in charge who hold up similar views to yours... except, they either go over what you'd consider an ideal decency level, and ban stuff that, to you, seems perfectly fine. Would you consider it a victory for decency, or would you be upset with the new admins taking things too far? What if they do it just for your liking, but someone else — not a small group of users — would count it as an overkill?
Sure, it's their ball, so they get to call what's good and what's not. That's what they're already doing. It may seem like law-abiding yet complacent indifference, but it's one of the choices the execs have made about the structure of the platform. Whether you consider it a righteous choice or an application of misunderstood libertarian ideals from days gone by, that's the ground state of the platform as long as the execs are fine with it.
With that in mind, why are you not playing by their rules? Why do you think your vision of their structure of the platform is better than that of the people in charge?
I think I can anticipate some of your response. I think you're going to say something about how there's an appropriate behavior and there's an inappropriate behavior, and the platforms in question should keep the latter in check for the good of their users.
What if the chaotic nature of the platform is the goal? What if, again, the pictures of dead babies (as revolting as the thought is) is the price you pay for allowing for creativity and discussion to thrive? It's not a coincidence that it was on Reddit where /r/wholesomememes or /r/AskHistorians appeared. The same place that gives space to the far-right activists who can't withstand a kernel of criticism also gave us a community of some of the nicest people on the Internet, as well as a notoriously moderated place to ask serious historical questions and expect a thorough, scientific reply. Each found themselves amidst the chaos because they had the conditions to grow and thrive as they are.
Again, I don't think it's as simple a question as "Shall we ban Nazis?". I don't think it's even about Nazis. As appealing as they are a target (and sure as hell an easy one), it's a conversation about regulations, and regulations are a matter of opinion. One could make a case for said dead baby pictures being an invaluable asset in one's drawer (I'm not going to, but one could). As far as I'm aware, Google, Reddit et al. already begrudgingly obey the law, and that's as much as one can legally ask. Beyond that, the ethics of allowed content become a matter of preference.
These sites need to enforce decency
No, they don't. They could, and many feel that they should, but they work well enough as it is, and by giving moral authority to one source, they inevitably upset everybody else. As far as their business model remains "more views equals more profit", they're doing just fine in status quo.