The other way around. SpaceX benefits from modern technology and modern manufacturing capacity, which make scaling things much easier than during the Cold War. Overall production has grown greatly in most fields, and top companies often rival superpowers of the past in their field.
Labor cost have gone up not down. Rocket manufactre is still incredible labor intensive. The Soviet Union had very low labor cost. The only way SpaceX could do it is by innovating into renewables.
The Soviets did it mostly be mass manufacture of 1960 technology and most launches still today use the same tech.
