I wouldn't say it was a foolish insistence and in many way letting Russia launch the first satellite was a brilliant move on the part of the Eisenhower administration. The most significant thing is that by having Russia send a satellite over America, it established the overflight principle that low-Earth orbit was a separate domain in international law similar to international waters in the oceans.
A legitimate concern was that if America sent a satellite up into space, that any time it traveled over the Soviet Union that it would be treated as invading Soviet airspace. In theory that could be considered a casus belli for some sort of significant response that would provoke military action.
You could say ditto for even sending a crew member in orbit.
As a result of the Soviet Union sending the first satellite and then sending Yuri Gagarin over the USA at orbital altitudes, the USSR had no justification and reason to be objecting if the USA did the same thing over the USSR. IMHO that was utterly brilliant.... and at the same time making the USSR prance around like some sort of victory was achieved when in fact they gave up a major diplomatic point of order in international law. It really didn't cost the USA much of anything other than temporary prestige that is largely irrelevant today... and was completely made up for anyway with Neil Armstrong's landing on the Moon.
The goal of the Eisenhower administration was to send spy satellites over the USSR, something that happened not too much longer after Sputnik. Unlike what happened with Gary Powers and the U-2 plane getting shot down over the USSR, they had no reason to complain about satellites.
How is that foolish that Eisenhower waited to have the Explorer satellite launch few weeks after Sputnik?