There are other reasons given for a woman not being able to be a rabbi, but I can’t remember whether they’re entirely disconnected from this or based on it…
R’ Moshe writes that a woman being a Mashgiach for a kashrus organization is a problem of serarah, unless she only acts as the eyes of the Mashgiach and does not enforce the rules herself (and even then, it’s possible that that would only be good enough in the bedieved case R’ Moshe was dealing with). It’s hard for me to see how being a Rabbi or Rebbi should be any better.
It’s interesting that women are totally in charge of the kashrus of their homes and Taharas HaMishpacha, but are not favored to routinely be mashgichos.