The US Moon Landing was only the biggest win because you were conditioned to see it that way in the western school system. Russia had claimed many firsts:
- First intercontinental ballistic missile and orbital launch vehicle
- First satellite
- First person in space
- First robotic space rover on the Moon
- First probes sent to Venus and Mars
Considering these firsts, suggesting that Russia should have focused on being the first to send people to the Moon to claim that just seems arbitrary. Apollo was highly criticized because there was nothing a person could do on the Moon that a robot couldn't do, and that NASA was unnecessarily endangering the lives of people for national prestige.
From
The Atlantic: "Over the years, I’ve spoken with many people who think deeply about space travel, and when I ask some of them about the whys, they admit, a little sheepishly, that there might be no compelling reason to send people into space—robots, yes, but people, maybe not. They seem hesitant to even say it aloud, as if to do so were blasphemous."
There wasn't actually a scientific reason to send people to the Moon. Why should Russia give importance to something the US arbitrarily decided to do?