News organizations get paid by the pageview, so they have an incentive to sell a story, not spread the truth.

Harper's magazine is not a website that counts pageviews as it's prime metric. It makes money via subscriptions. Different business model.

In fact, I think it might be interesting to start a "meta-journalism" organization that would find big stories in the media, talk to the people who were interviewed, and get direct quotes from them on if/how they were misrepresented.

That could be useful for giving people a better idea of how the media works.

You hire a publicist or PR firm that does the reverse and takes your story to journalists and makes sure they present it accurately.

That's a naive view. There no way a PR firm can force accurate representation.