There was indeed a need for science and engineering excellence after Sputnik was launched and Ike seemed completely caught off guard--I believe he was playing golf at the Augusta National in October of 1957 when the news came in. Also Ike was roundly criticized for his projects--the vision of an interstate highway system (modeled after the autobahns he saw in Germany). I was a kid then, but I distinctly remember relatives claiming it was an intrusion into state's rights to build their own roads and charge tolls (although we did have some U.S. highways, Rt. 66, for instance). Not wanting to antagonize Mr. K, he also did not lift a finger to help Hungary in its independence move against the Soviets ... and he did little more than referee the Suez Crisis, or manage (when he could help the French and quickly snuff it out) the growing threat of communism in S.E. Asia...to me he was acting in some of the finest liberal traditions; hardly what you'd call Reaganesque. In saying that, however, I still remember him as being a pretty good president.
Sputnik was another 'event' that was way overblown. We and the Russians were both working toward putting something in orbit. The Russians looking for propaganda managed to orbit a 2 pound grapefruit sized radio transmitter that could only say beep, beep, beep. I remember listening to it as it passed over on my older brothers SW radio.
A few months later, (and after a few spectacular launch pad failures which the Russians would never show) we orbited Explorer I, a 30 pound payload that actually did something other than beep. It measured cosmic rays and confirmed the existence of the Van Allen belt. It built by the JPL and was way ahead of anything the Russians had.
The Soviets were into stunts and the media always loves stunts.
I suppose the media reaction drove Congress to take control away from the military (the Army and Navy both had their parallel programs then) and create NASA to coordinate space research and development. We can argue if that is a good or bad thing but in reality, there was no science gap. It was all knee jerk reaction.