I expect that the amount of "good influence" institutional shareholders can exert on SpaceX leadership and operations is about zero, and the amount of "bad influence" is more than that. Thus, the only way this can affect SpaceX's leadership is negative.

A big part of how SpaceX did what they did is that they weren't beholden to institutional pressures. They could afford to take major risks. This may change when a pool of investors who don't care about space and just want the line to go up end up being stakeholders.