The US was clearly seeing private companies superior to socialized government solutions. The Soviet way was bad. The US way was good. But as Soviet were far ahead of the US in the space race, the US decided to put a lot of tax money into beating the Soviets by going to the moon. But how was this idea presented? A large government institution costing billions of tax dollars, wasn’t that very close to “communism”? How did the US “package” that idea to appeal to the public who hated socialism?
Mainly the Sputnik project freaked the Hell out of a lot of people. If the Soviets mastered space, the USA would be completely defenseless from that front, so the Space-Race became primarily about military technology. You'll note the the early astronauts were all Air Force personnel. The rockets used for NASA were research for better ICBM missiles.
Also worth noting that these programs were funded under mainly Democrat Party governance, who were far more receptive to Socialist methods and government intervention than were Republicans. In the current era, the more "American" approach has taken hold, with private companies like SpaceX and Virgin leading the way in space technology.