I think it is worth considering that demonstrations - a constitutional right - vary in terms of legitimacy; that is, they can be placed along a continuum of merit. Some demonstrations do a whole lot to advance critical thinking and progressive thought, while some do not. I think it is dangerous to be dismissive in regards to the "less legitimate" demonstrations, as I believe they add depth to - and can actually segue to - "more legitimate" ones, contributing to the democratic process broadly.
I definitely agree there is a continuum, and that protests should aim for tactics that best facilitate broader social discussion on the issue. And I agree with you, as well, that even those protests which fail to do this can still contribute more broadly to the democratic process. I should clarify, that when I spoke of 'illegitimate' protesters in my post, the example I had in mind concerned the protests following last year's Trayvon Martin shooting - I read one account of a (white) participant shouting 'we want war' and then having to be talked down by other protesters, who more wanted to bring attention to the issue in a non-violent way. So, while I don't think there are too many 'illegitimate' protests, I do think there can be 'illegitimate' protesters, who might use protests as an excuse to make trouble and don't actually care about the issue-at-hand all that much.