It's not an exaggeration for some fields of work. While many people do indeed exaggerate the technological marvels of the Third Reich, there is no "marvel" here, just the fact that the Germans had focused their research energies and monies on different topics than the Allies did, and so the Allies were happy to loot their brainpower and research when it came to the Cold War. There are several areas in particular where this happened — rocketry is perhaps the most famous, but there were others (US Cold War submarines benefited immensely from German developments, for example). As a marvel-puncturing maneuver, it is worth noting that the V-1 and V-2 rockets were not effective weapons for World War II and so the German technological "bets" did not in fact pay off in the short term (the missiles were too inaccurate and too expensive in their initial form, and had no effect on the tide of the war, and arguably if they had spent those Reichmarks elsewhere it might have had a more meaningful impact). But combining that technology with the promise of nuclear weapons (which compensated for the accuracy problems and increased the damage potential of any particular rocket) was very appealing for the Cold War powers.

Early US ballistic and cruise missile work in the Cold War were literally clones of German technology. The V-1 became the Loon missile, which itself birthed the "next generation" of cruise missiles (e.g. Matador, Regulus), and the V-2 became the initial blueprint for US ballistic work as well (e.g., the Redstone missile). Individual Germans played large roles in the US program. Von Braun is of course the best known but there were many others as well (there were Germans working throughout the US rocket development programs).

"How important was X to Y" is a difficult question to ask of any phenomena, because it is implicitly a counterfactual ("If X hadn't been there, would Y have been different?"), and counterfactuals are hard to answer. But we can certainly say that the Americans thought the German technology was the "next generation" of work and that they desperately aped it, employed its creators (and shielded them from political consequences of their having collaborated with the Nazis), and considered them to be more than mere technicians. (The Soviets also used Germans but tended to put them in more technician/engineer roles as opposed to management roles.) If the question is, "was the German technology a major stepping-stone for the US work?" the answer is unquestionably "yes." If the question is, "would the US have been able to develop all of those things without any Germans?" I think it is a harder thing to answer (and would involve follow up questions about how much time would be added or removed, etc.).

Of course, at some point they moved beyond the initial German tech and even participants — the Regulus was not a V-1, to make one example, even if it was directly descended from it — but that doesn't remove the fact that the German work had a huge influence on its development.

There are many academic works on the influence of the German technology on the postwar American rocket program. Though as a book it has its flaws, the introductory chapters to Walter McDougall's The Heavens and the Earth: A Political History of the Space Age (Basic Books, 1985) are largely about the German influences on both the American and Soviet programs.