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-- However, the person filing the suit would have to have "standing", i.e., he would have to demonstrate that he was personally harmed by the specific act of criminal violence (e.g., the victim or the victim's parent.) --

Inciting violence is a criminal act, charged by the state if it chooses to.

Not saying Sharpton or Jackson meet the criteria for the crime, but if they do, standing isn't the problem. The legal hurdle is 1st amendment, and that the person has to have expressly "urged ten or more persons to engage in tumultuous and violent conduct of a kind likely to create public alarm." (cribbing from NYS Penal Law 240.08, because that came up near the top of the web search results). In New York, the crime is a misdemeanor.

I agree with your comments about the difficulties of mounting a civil case.