Blackrock Vanguard et al have tremendous influence as shareholders even if it's via an index fund... see what they did to Exxon Mobil a few weeks ago!

Exactly right - BlackRock and other large indexers have no legal obligation to follow the recommendation of an independent adviser (like ISS, Glass Lewis) in a proxy contest, even for a passive fund. In most cases they passively anticipate the index provider's (S&P, Russell, MSCI, etc.) moves for corporate actions, but management elections for the board and executives are much more subjective as they don't influence the position itself (note that the manager can sometimes also be the index provider, like BlackRock = iShares). The smaller index managers almost exclusively follow the proxy adviser's recommendation because they don't have the resources to analyze board decisions for ~8,000 different companies.

That's a necessary point, but who really has the power? What is the chance that the investors will pull out, especially if they are making a profit?

Vanguard in particular is just a bunch of different mutual funds and ETFs. It doesn't do hedge fund type investing.