There are a fair number of engineers at centers (Stennis, Ames, Kennedy, etc.) that are government employees. When I was NASA-adjacent, it seemed they wrote the specs and testing regimes. I think the government even did some of the testing with government-employed test engineers and technicians. But yeah, a lot of the manufacturing is done by contractors.

There's a joke in the aero world that F-16s are designed by people Ph.D.'s, manufactured by people with Masters degrees, flown by people with a Batchelor's degree in History and maintained by people with a High School Diploma.

It turns out you have to make jobs for people at all levels of education and experience.