Er... no they didn't. They just were far more willing to take chances in order to beat the US to some goal only a matter of weeks or months before they knew the US would do the same thing. The entire 'missile gap' and technology gap were part of a propaganda campaign by the Soviets and abetted by the US Areo-Sapce industry looking for ever larger government contracts.
Good point. The Soviet “lead” was never that great, and we soon surpassed them. But Democrats used the opportunity to bash the Eisenhower-Nixon administration as being unprepared and behind the times. In other words they didn’t let a crisis go to waste, even back then.
Now for our 2011 Sputnik moment, whatever that means, there will be no bold new space initiatives and as the shuttle program winds down, tens of thousands will lose their jobs.