The Space Race wasn't about science, it was about getting your missiles into earth orbit first and claiming military superiority.
If the Soviets lie and say "space travel is possible, and we were able to get rockets there," why would the Americans then admit "actually, we found out that space travel does not seem feasible" in contradiction to popular science which had been postulating that space travel should be possible for hundreds of years? That is tactically disadvantageous.
The Space Race started immediately after WWII. The competition was raging for 12 years from 1945 to 1957, with many failed rocket launches on both sides. Is it a coincidence that the US happened to achieve success with the Explorer I within just three months after Sputnik?
The missiles are not in orbit, Tom! ICBMs never go into orbit! They are just extremely long ranged missiles!
That aside, Explorer 1 could have BEATEN sputnik if we weren't being morally righteous. We wanted to use a civilian rocket, the Juno 1, which was a derivative of the Redstone. We didn't want to use the Jupiter ICBM family due to it being developed chiefly by Werner Von Braun, a Nazi scientist.
Is it a coincidence that others developed planes right after the Wright Brothers' first flight? Is it a coincidence that any technology that is being vigorously pursued can happen several times, with only the first one being remembered?
There IS NO MILITARY SUPERIORITY in space. ICBMs do not give superiority, they end the world.
And ICBMs don't go into space until launched! They aren't in orbit! Nobody is trying to "get your missiles into earth orbit first" Until they are actually being USED to launch nukes they never move from their silo!