HugMonster - the last pilot that I can think of that made a decent job of running an airline was Sir Adam Thomson. On the other hand, I can think of many that thought they knew how to do it - and failed dismally.
To do as you suggest would be like getting a train driver to run Railtrack or a checkout person Tescos.
I agree though that lawyers and accountants shouldn't run them either. The best people to do it are the marketers - they, after all, are the people at the real sharp end: those going out and getting the business that pays everyone's salary.