The problem is that part of their value prop to customers is that they offer airport pickup (they send a black car to bring you from the airport to the rental agency). This puts them in a position where the airport has a right to demand use fees, since they are using the same roads and infrastructure as any other rental car service.
Reducing costs by not using the infrastructure, and therefore completely circumventing the need to pay the use fees, would be disruptive. Using the infrastructure but refusing to pay is not disruption. They are not doing anything more efficiently, they are just refusing to pay.
This issue should be understood as completely distinct from their core car-sharing business model, which is disruptive to the rental car industry.
The point is that FlightCar is offering the same or competing services as an Airport rental car agency without paying the same use fees as a rental car agency. This has less to do with efficiency and more to do with them exploiting loopholes to avoid paying the same costs as more established rental companies. They benefit from the increased rental demand, cab stands, and private roads produced by the airport, but don't pay for it.
If their entire business model is dependent on not paying the airport fees, it's probably not as disruptive as they claim.