Order-handling companies pay for "dumb" flow. Vanguard can reduce their outright trading costs to negative by being as dumb about it as possible, and then use these negative costs to artificially lower their reported fees. Just because Vanguard claims to be smart about it, doesn't mean necessarily they actually are incentivized to be smart about it or actually are in practice. People can still judge them by how close…
The company managing the Vanguard fund is owned by the fund holders themselves. It's a closed system and the incentives are aligned.
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As for proxy voters choosing the management, proxy vote research usually doesn't make sense. You see strategies amongst large hedgefunds like this: keep their valuable proxy vote research private, make a vote based on the most predictable-to-them-but-not-to-you outcome based on the research (positive or negative for the company, doesn't matter unless it will be picked up by others immediately, even then they can do things like exiting their position through an obscure hedge), and then hold or sell their shares based on the overall vote outcome and its implications in the research.
Researching and making proxy votes out of naive benign interest of the company is often just doing altruistic work for a greater collective, something markets frown on and usually punish.