This is an interesting question.
Seems to me that the AMA clubs promote the AMA insurance. All you would need is your own non sanctioned AMA club and appropriate liabilty insurance. I doubt it would be cheaper then what the AMA offers but I think that's all you need. I fly on private sites all the time and quite honestly all I would need is liabilty protection. Why does the organization need to be non-profit? Why not open it up for profit and get several insurance vendors involved. I think that's a great way to get better service out of the organization.

Now before I get flamed or you respond with PO'd smileys in your post - keep in mind this open conversation and we are mearly discussing the what if's. Ain't nothing wrong with that - but there is pleanty wrong with burying your head in the sand.
RickP