I'm not sure the USSR space race is the best example to use to make your point. The largely accepted reason as to why the US eventually beat them to the moon was that the NASA let a significantly more coordinated and efficient effort, combined with more robust and quality controlled manufacturing.
The USSR program had a number of competing chief designers ultimately promising ambitious, but commonly uncompleted, projects. They were known for treating each project as a standalone, as opposed to efficiently using each project as a stepping stone towards achieving the next. Their space program was 100% limited by the systems they had in place.
All in all, the "space race" was a marketing campaign on both sides to get the public to accept programs that honestly were too expensive for either side. After the whole moon race, both sides decided to change direction, got proper expectations in place, and ended up completing many other firsts that are a whole lot less sensationalized.
