Google is not digging up any information that hasn't already been made public. Any news stories it surfaces about an individual are part of the public historical record.
There are a lot of things that are made public that aren’t news stories. Should revenge porn be part of “the public historical record”? Can I publish all your private info on a random website and then claim it’s “part of the public historical record”?
If any part of the site is meant to assassinate my character, I could sue you for defamation in the necessary jurisdiction. Alternatively, I could work to improve the SEO scores of my personal and professional pages, acknowledge that you are attempting to character assassinate me in my pages, and move on with my life. The problem is that in reality what's happening is that people who made stupid mistakes as teenagers are now realizing that their arrest record is searchable, and I feel terrible for them, but wouldn't it be better dealt with by passing laws to thwart the arrest record sites rather than forcing every site that indexes content to delist on a case-by-case basis.
You keep harping on and on and on in this discussion about "private" information. What you are referring to are unauthorized disclosures. That is not the issue here, and I don't think anyone has a problem with removing that. Please stop tossing out this irrelevant strawman.
The issue is redaction of authorized, public information from the index.