I'll chime in on this lane swerve...Faculty Advisor is not a Gamma Phi Beta position; it's not a position title we use. We use Chapter Advisor and then we have various advisors to exec./other elected positions, if enough alumnae are available to serve.
However, Faculty Advisor is an official role that my university requires for all recognized campus organizations. In my case, I do that role on top of my internal GPB advisory board position. I have to sign off when the chapter wants to reserve campus facilities for events, I am sent their grades each semester, and I'm supposed to make sure they're following campus policies. I have a faculty friend who is a Delta Zeta, and she is their Faculty Advisor. Neither of us is THE main, internal Chapter Advisor.
I don't think the university cares about the affiliation of the Faculty Advisor...they just want someone on staff to oversee the groups/be the point person. There are probably universities that don't even require this role, but I don't know. I would think that GLOs would want a member serving in this role whenever possible. As far as official GPB advisors go, they must be members, and members who initiated as undergraduates or through AI are equal in status, rights, etc. There is no differentiation once you're initiated. And, only initiated members are allowed to wear our badge...and that swerves us back to the dilemma....