That's the thing, they weren't "the first with some early steps", they were way ahead of the US on literally every single measure that matters. The only thing the US managed to do first was land people on the moon, which is mostly a symbolic gesture.

However, because of a decades long PR campaign, people think the US "won" and don't even realize all of the USSR's achievements. That's the rewriting of history that I'm referring to. I'm happy to acknowledge NASA's achievements, as long as it's not at the expense of others.