For the first time in my life, I can point someone directly to useful academic writing on the exact real-world topic being discussed:
Appel, I; Gormley, T; and Keim, D, “Passive Investors, Not Passive Owners”, Journal of Financial Economics (Forthcoming),
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2475150If you don't want to read it: the evidence so far indicates that index investors are willing to bang the drum for better corporate governance. In fact, the authors argue that index-fund investors lead to "more independent directors, removal of takeover defenses, and more equal voting rights".
I cannot quite explain how satisfied I am to be able to share this incredibly specific nugget of information. It came up on a risk management module for an academic qualification I'm working towards, and I decided to pull on the exact thread the OP describes to see where it leads. It's by no means a settled situation at this stage, but things look reasonably positive so far.