Remember that space travel was birthed from the space race. America's main desire to get into earth orbit was so that they could send rockets up there and threaten the world into submission with ICBMs.
The US attempted and failed at building a rocket capable of getting into earth orbit for a long time. When Russia started claiming that they could send rockets into orbit, America's only choice was to claim that they could as well. After WWII America emerged as the world's biggest superpower. Do you really think 1950's America, a rich and paranoid government of the atomic era, would let an aggressor like Russia hold the hand of God over on them with ICBMs?
An entirely new, never before seen technology which does something physically impossible is something that all the money and power in the world simply cannot buy. America already had rockets that could launch up into the air until they were out of sight, and come back down unseen and the government did not shy away from deception.
Consider the
secret war of the Vietnam era in which the public and Congress were systematically lied to about the scale of the war and reckless disregard for civilian casualties for over 20 years between 1945 to 1967. The Secret War was administrated by the Eisenhower, Kennedy and Johnson administration, the same people who were in charge around the time when NASA started making all of its fantastic claims.
From the wiki link --
President Johnson had decided to expand the war while promising "we seek no wider war" during his 1964 presidential campaign,[8] including plans to bomb North Vietnam well before the 1964 Election. President Johnson had been outspoken against doing so during the election and claimed that his opponent Barry Goldwater was the one that wanted to bomb North Vietnam.