Why? "Owned by Vanguard" is the closest possible thing to "owned by the American public". What do you think Vanguard is and where do you think their money comes from?
Maybe just restrict index funds from interfering with whatever business they own? Index funds are ultimately a dumb momentum-based business model that just buys whatever companies are doing well at the moment and sells whichever aren't, but bear no risk themselves outside the economy shrinking. Not sure why they should have any say in how the companies they own operate, it seems like a complete competence mismatch (dumb ones with little risk controlling the smart ones with the skin in the game).