Of course companies have the rights to remove and restrict content on their platforms! I'm not saying that's a problem.

I'm talking about e.g. the DMCA requirements than an OSP be responsible for complying with takedown notices. This is not the company deciding to take action on certain content, but the company being forced to take action in response to certain submissions. The law pushes the burden of enforcement onto a private company.

As someone else posted in the comments, there are services that will go around and make (bogus) claims on your behalf. It seems plausible that's what happened to the author's original article, rather than that bing and google themselves were trying to influence the net neutrality conversations.