While I think this is a cool concept, I tend to agree with others here that the logistics/money would not make this practical.
If we are to assume this is a nationwide network in the US, this would take a lot of cash to buy up even small parcels of land to have these bicyclist campgrounds. And even with the campgrounds being low impact, there are maintenance fees. And if it was a membership-based organization, how big of an organization would it have to be to make it feasible, and how much would dues be?
Let's take ACA as an example: in 2011, there are 43,550 members. If this non-profit organization were to be as big as Adventure Cycling, and membership dues were $40/year which is ACA's base member rate, this would mean $1,742,000 in yearly income for the organization. There would most likely be other revenue streams etc, but how far would about two million dollars a year go towards acquiring land, building campsites, maintaining them, etc?
This concept might work better on a smaller scale, local level, and partnered up with a like-minded organizaton. I know in Mt. Hood National Forest a group set up a series of "Bike Huts" for mountain bikers.
I think the best idea is encourage those organizations that are already set up for camping, like state parks, to develop hiker/biker sites either in already existing campgrounds or in other lands they own. Or get more of these small towns along ACA routes and in popular touring to allow free/nominal fee camping in their town parks.
I also agree that warmshowers is a great resource and the fear of trusting strangers is overblown. We stayed with quite a few warmshowers hosts on our trip last year, and most hosts were good to great. We had a couple less-than-stellar stays, but nothing along the line of fearing for our safety.