You are correct, there are no contract red air fast jet companies that train pilots from day 1. However, the number of pilots required for contract red air is probably not as many as you or I may think.
Equally, there are quite a number of pilots that have recently left or about to leave that are more than capable of filling these seats that the Service is losing anyway. This, coupled with the amount of money and assets thrown at providing Red Air I am sure it makes more economical sense to have contract Red Air.