The soviet union was the first one to go into space and had far more of the outer space firsts than the US. They also had no market incentives to begin such projects.
It's ridiculous you could even bring up the cold war without acknowledging the major scientific and engineering breakthroughs done by communist societies.
Both the US and Soviet Union basically stole rocket tech and rocket scientists from Germany after WW2. The only reason the US and soviet governments supported and funded those rocketry efforts was due to the military and intelligence benefits.
There is 50+ years of evidence that we will not become multi-planetary without private funding. The only reason NASA was funded enough to get to the moon in the first place was because of the cold war. Very unlikely it would have happened otherwise.