The Soviets scrapped their N1 after multiple launch failures & then pretended they never were in the race. You can argue if US won, but the Soviets definitely lost the race.
Crewed landing is a massive achievement, but it being the finish line of the race is the US framing, primarily due to getting beaten in all other firsts.
They did start on many LEO space stations that the US had no good answer to, but its questionable if this was still the same race or a race at all.
Many people expected the race to just go on - lunar bases, Mars, etc. Given that nothing like that really happened post US landing on the moon, one could really say the race was over the final victory belonged to the US.