No, the heads of federal agencies are usually external so they can be held accountable for fuckups independently of the agencies’ cultures.
The current head under Biden is Billy Nolen, a former pilot who flew the Boeing 757 and 767 and worked as an airline executive for years before joining the FAA [2].
By credentials, these two are definitely qualified.
1- https://trumpwhitehouse.archives.gov/presidential-actions/pr...
2- https://www.aerotime.aero/articles/30600-safety-professional...
My follow-up point was that experience in regulating airlines is more important than experience running an airline when it comes to leading the FAA. However, those who regulate airlines as a career are unlikely to be big political donors, so they're unlikely to be chosen to lead the FAA.