Osama bin Laden's $30MM (per Wikipedia) inheritance from his estranged family no doubt greased some skids, but mission-driven people with charisma come from all socioeconomic backgrounds, and some of them succeed. (FWIW he did not attend Harvard, and AFAIR never set foot in the US.)
The general pattern is that the wealthy are, on average, supportive of the existing order. They are the winners of the current game. Of course there are always rebellious children who are motivated by religion or power or fame etc, but most of them are feckless due to their upbringing. And ultimately, there are so few of them.
Leadership abilities can be found across all economic strata. Most are happy to leverage their talents into moderate economic advantage, but some are driven by "larger" causes.
I would say that the intersection of leadership abilities, belief in a "larger" cause, belief in victimization, and a willingness to harm innocent people (or to blame them for their inaction against your oppressor) ... is what leads to the risk of violence against the existing order.