"It would be better for everyone if we had this disagreement in public. -- Nobody, ever"

I disagree. I am hugely glad on both eBay and travel sites when I can see a scathing review from one person and a response from a seller or hotelier like "you tried to pay with monopoly money and you tried to return a bag of rocks"

Private responses are an acceptable alternative to public responses. They are not a substitute, especially if it leaves the original negative review standing in public.