Can and is it beneficial to deregulate the FAA (privatise it)?

Is it possible to write a bill which says that the FAA should be privatized but government officials should be put on its board of directors, so that a) they can vote against any uneven decisions that private stakeholders may take, and b) can continue to put forth some of the regulations. What the privitisation of FAA would do, is that it will driven by the profit motive, and it will gain revenue by charging the airlines who are part of their body. Thus it will be motivated to allow more smaller airline companies into the system as well as regulate the system to avoid too big monopolies or oligopolies. More smaller companies in turn would increase competitiveness as well as decrease ticket prices for consumers. Furthermore a privatised FAA can work closely with a privatised and modernised ATC system as well. Thus, essentially does this make sense or if not what else can a Republican Senator do, in order to keep the FAA deruglated?