Maybe contacting the author, first, was a better way to handle the situation?
Of course, the article author doesn't try to search her own flaws in her approach.
I disagree. Statements made in a public, academic context should be debated and corrected in a public, academic context. This is perfectly appropriate; indeed, taking it private may well be inappropriate. This is the nature of academia, and anything less is a disservice to the greater academic community in question, who are shut out of a private conversation.
Obviously some professional courtesy is called for; such debate should not be acrimonious, and sending a note to the originator of the debated items is probably appropriate unless you know they can't miss it. But I'm not seeing anything egregiously wrong with the author's approach or tone in this article.
Many of the corrections in question involve objective facts, too. While I am keeping in mind I am seeing only one side of the story, I have a hard time seeing what justifies responding to a debate about what the objective fact of the matter is with any sort of accusations or stonewalling. The March of Dimes example seems pretty open-and-shut, for instance.