Hi, Rachel! I feel your pain! For many years before retirement, I was the director/curator of one of only two museums in the country professionally managed on high school campuses. We had a board of the "Friends" of the museum. There had been, for many years before my arrival, a board that thought it was a true governing board, but for reasons way too complex and too long to include here (but including malfeasance), it was disbanded by the Head of School (the equivalent of a college President). For reasons that seem to parellel yours, I, too, tried to establish a faculty advisory group and had the exact same experience... in general, in that population, no one wants to take on more work, but they're happy to have their name listed (creds).
Don't assume that a community advisory board will be much different. Some will be devoted and want to help with real time, talent and treasure; most, not so much. Do you have a local cadre of alumni? Or retired former faculty? In my opinion and experience, this is the population most likely to truly help and may have (in the case of alumni) some sway with administration.
Finally, it will most likely be difficult to get admin to agree to "allow" an "outside" group to have governance authority. That's the difference between a museum that is free-standing, independent, and a 501(c)(3) corporation, and one existing within a "parent" organization. So this is the crux of the dilemma. "Board" members with no real authority, nor fiduciary responsibility are less likely to take risks to advocate for the museum, and institutional administration is not likely to be open to relinquishing "power."
Good luck with this. Happy to talk more off line. Vivian