No, that's not what that email meant.

Yes, the 3rd party providers get their contractual flying filled first, but for example RAH and Envoy aren't competiting for same flying that much. AA is the one who decides what gauge of airframe is needed for what routes. And since RAH doesn't have 76 seaters in DFW, then RAH doesn't get that flying.
Only direct competition in DFW is from Mesa, with some shuffling between Skywest, Mesa and Envoy. And lately Envoy has been getting good flying from those airlines.

It didn't mean regionals get to pick what they fly and then AA just has to sell that many seats.