It's not a straw man; it's exactly what's being proposed. This is an invented dichotomy. If a private entity is not liable for its users' content, it doesn't follow that they then can't moderate that content. We don't hold bars liable for defamatory statements their patrons make, even if they kick out people who say racist things.

If I understand correctly, you're also misapplying common carriage here. Common carrier laws basically say that if all you do is transport things from A to B, you can't discriminate. That already doesn't describe social networks, which do much more than just deliver your speech: they algorithmically promote it, sell ads against it, etc. But OP is speaking even more broadly than that. One example they've brought up is YouTube demonetizing people, which has nothing to do with transmitting speech.