It sounds a bit cartoon villainy, but honestly, I see no reason to doubt that he said this. Everything points to these people being casually desperate to be let into ever innermore circles. Even now that this particularly ugly circle is blown open, notice that they still simply do not talk about what their fellow insiders did except in vague generalities.
Back to your main point, mafia operates similarly. In fact, there is not much difference between the two. What is Larry Summers not saying there is that being part of this circle is making this circle more powerful. Them not talking about what they know is itself "influencing powerful people and outcomes".
Except for the parts involving criminal coverups. That seems to plague close-nit groups at any level of society, e.g. world religions, police, finance, families, etc.