I think we need to be a bit careful about painting this as public sector bad, private good, or vice versa..
Remember (?) NASA has rarely if ever built spacecraft….historically they set the project goals and then put production out to tender (hence the old lowest bid joke about Atlas/Mercury). The previously mentioned Surveyor was I think built by Hughes..who in turn sub-contracted out the sub-assemblies…
NASA retained the project management and it would be a mix of NASA and sub-contractor employees who would put the whole combinations of vehicles (launcher, payload etc) together at the Cape or wherever.
What’s changed in recent years is NASA pushing more and more of the management and coordination out to the private sector and concentrating on the science side with schemes such as CLPS, where they pay for payload carriage but increasingly leave it to the private sector as to how to get the payload where it needs to go.