Mike, I kinda spent all my elaboration in post #5, but I'll add two ideas.
First, what's that old advice about eating an elephant...take it one bite at a time? I think this is where the local chapters can play a significant role. If we're trying to pump up a national plan requiring choreography of students, instructors, the insurance industry and manufacturers, the first bite is hard to identify. If, on the other hand, we're talking about getting a machine available for training using third-party CFIs in my local area, I know where to start.
I believe if someone demonstrates the model locally, it will be easier to build a case for promotion of the model at a national level. If approval of LSA gyroplanes, success by Carter's PAV, or other unforeseen market force suddenly raises awareness of gyroplanes, that's a bonus, but I think we can start without it.
Second, we don't need to reinvent this wheel. This idea is used all the time in the fixed-wing world. I personally know the owner of a flight school which is also a Cessna Pilot Center (dealer). In at least two cases, he has had students buy their own Cessna 172s to use for their own instruction, then lease the planes back to him for use in training others. It's a win-win deal, and by keeping the airplane busy, it saves money all the way around.
Options for brand new two-place machines suitable for gyro training that would work in a lease-back situation are slim right now. SparrowHawks are not in production; the Sport Copter II won't be widely available anytime soon; RFD won't sell a tandem Dominator to a student; I'm not clear on the status of any new Xenons in the US.
But there are certainly underutilized machines which could be put into service. For example, there are two machines in the Pacific Northwest which might require some work to be airworthy, but are for sale at very low prices right now. The owner of one of them recently announced his willingness to take on a partner in lieu of an outright sale! There have to be other opportunities involving machines which guys have built and paid for, and are not flying enough to justify their fixed costs.
I don't know everything about the LODAs, but apparently a common qualification included is a minimum of 200 or 250 hours in gyroplanes to be eligible to train in the machines. This creates a barrier to entry by new CFIs using add-ons to their helicopter CFI ratings, but many experienced gyro pilots acquiring their CFI ratings would be qualified right out of the box.
Someone who subscribes to the AOPA's prepaid legal services plan might be able to probe the feasibility of these deals with a phone call, at no additional cost.