Flight Surgeon was a position within Starfleet's Fleet Operations Center on Earth. (ENT: "Demons" set artwork)
In January of 2155, Starfleet officers J.T. Tagamolila, J. Devlin, V. Fredricks, M. Kelly, and J. Billingsley worked as Flight Surgeons.
Flight Surgeon was seen on directory listings seen on the set of "Demons".
Flight Surgeon, like many of the other divisions, departments, or positions on the listing, is a reference to NASA and ISS terminology.