I suppose the reference is self serving in the sense that an oft-quoted statement is self-serving: "What's good for GM is good for the country." The
actual statement was "What's good for our country was good for General Motors and vice versa." (Source courtesy of Wikipedia)
We're all in this together, whether "this" is economic interests as with the former GM CEO or "this" is public health as with Vanguard here.
As to Vanguard's preference to work behind the scenes, it could have some validity but is somewhat suspect for a few reasons.
When countries boycotted South Africa in the
1980s, a major company that continued to do business there was
Coke. Another I recall was IBM, that said it was helping blacks economically with good jobs. No pun intended, but the question of how to address apartheid was not black and white. (The link is a pretty good presentation of the issues as seen at UCLA.)
A difference there was that what these companies that remained in South Africa were doing was visible and publicized. In contrast, Vanguard claims to be effective by working in secret.
Sometimes, perhaps too often, rational entreaties are dismissed unless there is a threat of force behind them. A case can be made that MLK was more effective because people saw Malcolm X as the alternative.
Vanguard, along with the mutual fund industry, is the 800 pound gorilla - that has the power to compel corporate changes. It may not be necessary for Vanguard to vote changes (though it would be nice, once in a while), but if they don't show that they are willing to use their ownership constructively, all their nice letter writing may go for naught.