No.
I'm saying:
(1) They aren't exempted from responsibility, full stop. Site owners and operators manifestly are held accountable for content by a variety of social mechanisms independent of legal liability; you are incorrect to equate responsibility or accountability with legal liability, which is a very small subset of accountability, and that alone, without any other errors, would be a fatal flaw in your argument.
(2) Also, yes, they aren't given the exemptions that apply to similarly situated parties in other media because the internet is fundamentally different mechanically or in ethical situation in a way which changes the ethical responsibilities of people actively deploying resources to selectively relay content, but because (in the somewhat idealistic view) because the perceived benefits of imposing the usual legal liability were viewed as less than the perceived collateral costs in economic development of freeing site operators from legal liability for user-supplied content with certain limitations, or (in the view more in tune with the realities of lawmaking) because the cost of buying off the objections of a set of big money internet interests with a liability exemption was viewed as outweighed by the benefits of imposing the sweeping obscenity and indecency controls they were packaged with.
> Now you're saying that they should regulate all content
No, I'm saying that operating a site inherently involves that, and that they have the right to choose how to do so, even when that excludes certain views from their site, and that that right is enshrined in the First Amendment.
I'm not saying anything about what they should do.