Yes, as I think you are also suggesting, even if in theory an institution would have a stake in speaking on a controversial issue, it might for practical reasons decide not to, or to very carefully limit what it does say.
To be very blunt, these institutions have long term strategies that depend on them being seen favorably by donors, well-positioned alums, sometimes government officials, present and potential faculty and students, and so on. And in fact different institutions might have different sorts of exposure, different branding strategies, and so on.
Of course this is the same sort of issue a lot of large for-profit companies face as well. “Public relations” turns out to be really complicated, because the public is complicated. And these successful organizations, whether in higher education or other competitive markets, did not get where they are by ignoring the importance of public relations.
But one of the complexities these higher education institutions face is maybe not so many people in the public are comfortable with such a blunt view. Personally, I am fine with it, but I think some people see it as crass and inconsistent with the ideals of higher education for these entities to treat PR issues in such a way.
Anyway, point being I basically understand where Roth is coming from. This is not always going to be easy. And while I think it is fine to try to articulate some general principles and rules of thumb, when things get hot, the top administrators at these institutions are going to have to make some complex decisions about what best balances the interests of the institution and all its various stakeholders.