They do that now. Most of the "real work" is done by contractors. Grumman built the Lunar Module. North American Rockwell did the Apollo CSM package. Rocketdyne, RCA, and others got pieces of the action. Private companies all. Universities also got a slice. The MIT instrument lab did quite a lot with the LM systems. JPL has a university connection.
And yet we have a space ship that can't survive foam insulation or rain.