FWIW,

here around Nashville, I've been told by the Parks & Rec directors in two separate counties that a city who manages a "public facility" can NOT, by law, require membership in any particular organization, club, or group...nor can they prohibit use of those facilities based on membership or non-membership in any such organization. So, they can not, technically, require AMA membership.

Now, what they CAN do is require proof of a certain amount of liability protection...and they may choose to accept (or not accept) whatever form of documentation from whatever source of said coverage they wish. So, they can certainly say "Your AMA card demonstrates proof of satisfactory insurance coverage." They also, as it happens, will accept homeowners or renters insurance with satisfactory documentation that the activity of flying RC Airplanes is specifically covered.

I freely admit, I have no idea if this is a local, state, or broader set of rules, nor have I done the research to know, specifically, which section of whose codes applies here.