Yes. Definitely there is some truth to that. There were plenty of aviation test programs that were kept as closely guarded secrets initially, but several of them became very well publicized for propaganda reasons - both domestic propaganda reasons as well as international.

As but one example (well highlighted in "The Right Stuff"), Chuck Yeager's first blast beyond the Mach 1 "sound barrier" in his X-1 rocket plane "Glamorous Glennis" in October of 1947 was initially kept under wraps and classified as Top Secret by the US military, but was quickly publicized just a few months later in 1948 in order to exploit the monumental aviation event for domestic PR reasons and international bragging rights. Likewise when Scott Crossfield surpassed Mach 2 aboard his Douglas D-558-II "Skyrocket" in November of 1953 (flying primarily under the civilian NACA - National Advisory Committee for Aeronautics - banner). The publicizing of the famous X-15 rocket plane flights also served immense propaganda value and many of those missions were publicized heavily afterwards, helping to raise American aviation prestige domestically and globally. Aviation accomplishments in those (and many other) prototype aircraft served as a demonstration of American USAF/military and civilian prowess to both allies and enemies like the Soviet Union.

From the end of WWII into the early 1960s was really a golden age of aviation, and there were plenty of unknowns about flight that needed to be resolved as technological advancements allowed for higher speeds and altitudes to be achieved. Attacking the transonic regimes at and above Mach 1, as well as high altitude flight profiles, offered immense scientific gains and new understandings of aircraft performance behaviors and flight characteristics, and the knowledge gained was of course exploited in both the military and civilian arenas. These test flights were designed to examine how best to understand and exploit various stages of flight, and were incredibly dangerous, with many countless test pilots losing their lives in a huge variety of bizarre aircraft as they "pushed the envelope" to see what was possible.

Several of these test pilots (such as Yeager and Crossfield, or guys like X-15 pilot Bob White) definitely became quite famous after setting aviation records, with their faces published in issues of LIFE Magazine and their stories told in books and newspapers in America and beyond. Likewise of course were the Project Mercury astronauts, who also saw a tremendous amount of fame for their risky efforts. Many of them became like the rock stars of the aviation world, easily recognized in America and in some cases even becoming famous globally. Countless other test pilots though were not nearly as high profile and never received anything close to the same level of attention, even though they took the same risks. It really was the record setters like Yeager and astronauts like the "Mercury 7" who got the vast bulk of public attention.

The US military and political apparatus recognized that a certain amount of publicity was required in order to goad the taxpayers into being willing to support very expensive endeavors on the table - particularly the civilian manned space programs like Mercury, Gemini, and Apollo. Walter Cronkite, when asked about the fame many test pilots and astronauts received, once summed the era up by saying "America needed new heroes", and test pilots and astronauts fit the bill, willingly risking (and some even losing) their lives in the pursuit of advancing human knowledge and understanding of aviation and spaceflight.

There was a famous (and very accurate) line highlighted in "The Right Stuff" when one of the Mercury astronauts, when talking about the need for the press to be allowed access to the civilian space program, stated that "no bucks, no Buck Rodgers" - meaning that in order to get the American taxpaying public to support and fund these programs (providing the "bucks"), that public would need to see and learn about the men taking the risks (the "Buck Rodgers") to appreciate where their tax dollars were going. The press, when allowed by the Powers That Be, did their part in publicizing these dangerous endeavors to whip up public support, using the idea of the Cold War "Space Race" against the Soviets to help fuel the "us against them" peaceful battle for the conquest of space. The astronauts (and their counterpart cosmonauts) essentially became the soldiers on the front lines of that peaceful Space Race, with their heroism and the idea of being strong, family-oriented role model patriots willing to risk everything for their nation being a storyline that was played up heavily by the media.

The launch of Sputnik 1 in October of 1957 really did not alarm the US government too greatly (or not as greatly as many may think or as was portrayed in "The Right Stuff"), as both Soviet and American success in launching a satellite was expected at some point in the near future, so for those in the know, the "Space Race" between the two nations was already clearly coming. Sputnik 1 however did dramatically influence the propagandizing of the aerospace industry and thrust the Space Race into the public fold very quickly, forcing the American public to recognize the threat potential that the idea of an artificial "Red Moon" from the USSR orbiting the Earth could present. Prior to Sputnik 1, many Americans felt that the USA would be the first to send a satellite onto orbit above the Earth, considering that to be essentially a given. For many American's, Sputnik 1 came out of nowhere and shocked the nation in that regard, suddenly forcing America to take a back seat to a massive Soviet communist achievement (and likewise with Yuri Gagarin's first flight in his Vostok 1 capsule in April of 1961), and that shock and fear the American public felt about the USSR suddenly being able to potentially dominate the true "high ground" of space was definitely capitalized on by the US political apparatus to increase funding and push the nation's public civilian space programs and far more covert military/intelligence aviation and space efforts forward at a much faster clip. The competition between the USA and USSR for space supremacy - the so-called "Space Race" - created a lot of public taxpayer support and thus allowed for massive budgetary increases in the aerospace industry to be approved by Congress, both for the civilian space efforts and to fuel the nation's military industrial complex.

Similar to the main question, how would USSR's knowledge of the flight impact the US, and it's space program as a whole?