Personally, I think there is a difference between publishing companies that develop their own content and social media companies that only provide a mechanism for others to communicate.

I'd count the latter among the infrastructure companies.

I wouldn't. Even just the design of the tools they provide to facilitate communication influence what content can be created and how it is created. How can Facebook be a common carrier when it has age restrictions on accounts and makes choices about what languages and scripts it supports? If it were a common carrier, it would be unable to filter or limit the dustribution of pornography, for example. Should the be coerced by legislation to drop all such restrictions? If they should then what about private forums and bulletin boards? They provide essentially the same kinds of service? I just don't see how that approach is at all tennable.

Every platform limits the content based on the design of the tools. Being a common carrier wouldn't mean Facebook would be forced to support particular scripts any more than the phone company was required to develop video calls.

If there are some restrictions we want as a society, like age or porn, those could be made exceptions.

Regarding private forums, I'd distinguish between forums that seek to serve 99% of people and those that serve less than 1%. The first are public; the latter are private.

Facebook is just the last platform in a stack of platforms people use to communicate. Why would Facebook be allowed to restrict content while those who provide the services Facebook is built on wouldn't?

I already explained exactly why. They already have responsibility for the content on their platform simply due to the nature and design of the platform.

ISPs also "have responsibility for the content on their platform" and police content already (for piracy, for example).

That's completely beside the point. They are required to do that by law, in exactly the same way that the phone company can be required by law to cut of service or trace calls by the law or the courts. Its got nothing to do with common carrier status, which is about the discretion of the company not the government.