Hmm. I generally disagree – in most cases, I would say say that it can and should be done privately as a first step. (Albeit sometimes with a request to apologize publicly.) Only in relatively severe cases is a public reprimand the ideal first step.
Community is a complex topic, and we’ve got a lot of complicating factors here. Yes, we want to be welcoming, and harsh measures are required if somebody gets directly in peoples’ faces. But humans are humans, and public confrontation will, more often than not, cause a situation to spiral for the worse, where a more private reprimand may quiet it down.
We have to remember that the goal is not slapping people down, it is to make the public community as friendly as we can. That being the case, I would generally save public reprimands for particularly bad offenses and second warnings, but start privately where possible.
(All that said, I would say that a less-specific “let’s cool this conversation down”, publicly but not aimed at anyone in particular, is often a good first step.)