The true story behind Sputnik is very different.
From http://www.trumanlibrary.org/museum/sputnik1.htm
“Perhaps the most shocking thing about Sputnik for U.S. scientists was its weight — 184 pounds. The U.S. had its own space program, but it was working on satellites that weighed a fraction of that.”
Fact is Sputnik went up on October 4, 1957.
Explorer 1 became America’s first satellite on January 31, 1958. (pretty quick response) Weight - 30.7 lb
However due to our use of less powerful rockets, the US was forced to utilize miniaturization. When it came to the moon shot, the Soviets did not upgrade their rockets like we did and could not get the weight down. That is why they never made it there.
The fact we finally overcame the Soviets had nothing to do with mobilizing educational resources, however, and everything to do with infusions of money and existing private sector intellectual capital. We were on the moon before any "sputnik inspired" engineers finished school.