The government was the customer, so it was up to them to decide what to do in house and what to hire out. They chose to manage the project in house, creating specific design requirements and contracting out work to meet those requirements, not to create the designs.
Private industry did do most of the work, but they had no leeway to influence the design. Even when they tried to make choices, e.g. Morton Thiokol's Engineers' no-go decision on the Challenger launch due to low temperatures, NASA administrators effectively overrode that decision, by directly calling unqualified employees at Morton Thiokol, until they found one they could pressure into giving a go.
Private industry did do most of the work, but they had no leeway to influence the design. Even when they tried to make choices, e.g. Morton Thiokol's Engineers' no-go decision on the Challenger launch due to low temperatures, NASA administrators effectively overrode that decision, by directly calling unqualified employees at Morton Thiokol, until they found one they could pressure into giving a go.