As a man/person I still wholeheartedly agree here. Ultimately it comes down to recognizing exactly what your end goal is in choosing one route over another. At best someone will ignore public criticizing and start thinking of the criticizer as a jerk, a perfectionist, a "nagger", or just annoying. At worst they take it personally, start avoiding work that might lead to public shaming, and start fearing parts of their work day.

Engineering is stressful enough. No need to drive people away faster than necessary. Private criticizing can be delivered directly to offending person, especially if you're a team lead or manager, or it can be delivered to your manager. If it's significant enough to result in actionable items then the situation will be handled one way or another. Meanwhile if the person is capable of resolving the issue after your private criticizing they'll often respect your decision to keep things professional. Heck, sometimes there's an entire bag of worms under the surface like serious family problems they're going through outside of work at that same time that explains their behavior, something you almost certainly wouldn't find out with public criticizing.

I've seen some significant turn arounds from previous team members after some pointed private talks. Alternatively, I've seen people dig in their heals and become a problem team member after improperly timed public shamings.