1) you're really just aiming to regulate the selective censorship (which is in a direct relation to the existing monopoly laws)
2) why limit it on the Internet? TV and newspapers played exactly the same game for decades and they still do on some subjects. And when no major newspapers mentions a certain story, it's pretty much to the same effect as if google's removed it from the index. You need to first hear about something to know what to google for, and if it's not on a few mainstream media sites for most of people it just didn't happen.
Google uses the lumen database for DMCAs, which account for most of its removals, I am not aware for something similar for non-dmca removals however.
"Zerohedge has been moved to page 10 for the following searches:" strikes me as expensive, impossible in the general case, and relatively useless.
I have no objection to Facebook, Twitter, Google/Youtube, Hacker News or WeChat aggressively moderating their platform, or even running social engineering programs. Their platform, their problem, humans have lived with that sort of thing for centuries. My issue is that content recommendation systems are not transparent about why things don't appear.
Zerohedge got booted from Twitter recently. I don't mind that sort of thing because it is obvious what happened; go to their twitter page and it says the account is suspended. If Google tried something similar and removed them from the index there isn't a tool (that I know of) saying "Zerohedge has been removed from Google's index" and that is upsetting. I'd support legislation where people had to provide that sort of information in a public and accessible manner.
I suppose my big complaint is that if the big companies were organised against a group (maybe Google has an irrational dislike of Russia-based IT companies; it is possible) then that group has no way of identifying that they are being treated differently. This will always be a concern for political minorities.