The AAM model is the only one I can see as a viable solution--otherwise, the admission is tied to having a certain product (a Bank of America card or AAA road services) that many people won't need or can't get. The other problem is, they're too restrictive--one weekend a month is not much, and AAA generally only offers small discounts, not free admission. Of course, that doesn't mean Bank of America couldn't run a subscription service only for museum admissions and not tied to their products or other restrictions, but I don't expect they'd be interested, it doesn't really fit well with their business model.

As far as the AAM or local museum councils go, I don't think the average museum visitor would want to join them. And I don't know that they would want or could handle a large influx of members who are only interested in the free museum visits and who aren't museum workers or professionals. However, I could see the AAM helping create a separate subscription service.
However, I think it would be easier just to create a new service, and take whatever help these organizations are willing to provide to getting it running. The funds would primarily come from subscription fees, but I have no idea if that would be enough to be viable.