I guess it depends on how you view work. I can dislike someone's work as a colleague, but like them as a person. And vice versa. Work is just work - it's not our entire life. And someone being bad at a job (even if we accept that this person is truly intrinsically incompetent, and not just a byproduct of a dysfunctional org, as is often the case) doesn't automatically mean, to me, that they have some personal moral failing or personality flaw.

So, in that vein, I think I'd hesitate to publicly embarrass someone merely for being bad at a job, since that crosses over to affecting their personal life. If someone asked me about that person in a professional context (to make a hiring decision, for example), I'd be frank about their weaknesses. But I don't think the whole world has to know about it.