"The USSR did beat the US to orbit (and in space station technology), but it fell short getting to the Moon, launching scientific space probes, and building a space plane."
Those are arguably issues of running out of resources, to some extent, not inability to innovate. At some point you need money to turn innovative ideas into reality.
Elon Musk got asked what’s the best way to get started on building a space oriented company and his response was essentially “build a billion dollar internet company first”.
By innovating and letting private individuals to do as they please in their own self pursuits the US was able to get that money for space.
Those are arguably issues of running out of resources, to some extent, not inability to innovate. At some point you need money to turn innovative ideas into reality.