"Winning the space race" is a rather woolly concept and depends on your definitions. Although NASA did land astronauts on the moon, the Soviets had firsts in most of the main areas relevant today (first satellite, first astronaut, first space station, first landing on another planet, etc etc.).
Don't disagree with you, but, so far, the US is the only country to land people on the moon - and they first pulled that feat off 55 years ago!
Of course, it's not clear whether they could pull that feat off right now, today, but they did pull it off and nobody else has. Suddenly nobody remembered the Soviet accomplishments of first satellite in space, first man in space and so forth. All they remember and care about is first man to the moon.
After all, America does a great job of marketing herself! :)
America landing on the moon signaled the end of the "space race". The soviets could have pushed more money into it and landed one of their cosmonauts on the moon, but they just gave up because to them "the race" was lost, and not worth putting so much more money to come in second-place. All their other "firsts" were like trial/qualifying runs for the big race of landing on the moon.
That's pretty much the way I understand it. I'm sure the Soviets could have landed a cosmonaut on the moon in the early-to-mid 70s, but why bother? It would actually look lame! Hey! We can put men on the moon, too!
There is an effort to put the first woman on the moon, but as far as I can tell, the US is the only country concerned with that. Too bad, because that title has been there for the taking for over 50 years now.
The key is to pick a winning criteria that is sufficiently meaningful looking and then really sell the heck out of it.
There were lots of good arguments for the Soviet space program having been ahead, lots of landmarks they hit first. But, ya know, the dude standing on the moon couldn’t hear these arguments.
1st to anything is considered a significant metric, it's simply picking the larger tasks and working your way down to possibility to find effective goals.
https://en.m.wikipedia.org/wiki/Timeline_of_the_Space_Race