For the full quote:
"the Russians were so thoroughly stood down, and we knew it. They didn't make any move. They did not increase their alert; they did not increase any flights, or their air defense posture. They didn't do a thing, they froze in place. We were never further from nuclear war than at the time of Cuba, never further."
Source is listed as: Kohn, R. H.; Harahan, J. P. (1988). "U.S. Strategic Air Power, 1948-1962: Excerpts from an Interview with Generals Curtis E. LeMay, Leon W. Johnson, David A. Burchinal, and Jack J. Catton". International Security 12 (4): 78–95.
Is the extent of the crisis exaggerated?
Probably not.
First, it might helpful to have some information about the person who gave the quote you cited. General Curtis LeMay was head of the U.S. Strategic Air Command during the crisis. Certainly he was well informed. However, it's also fair to say that LeMay was unusually bullish on the prospect of nuclear war.
This page quotes extensively from Richard Rhodes book Dark Sun, which is a history of the development of the thermonuclear bomb, and peripherally of the evolution of the cold war. The book talks a lot about LeMay.
In particular, LeMay routinely ordered his reconnaissance planes to overfly the Soviet Union. This was done in violation of both Presidential orders and international law. The nominal point was to map the U.S.S.R., but LeMay explicitly stated that a secondary goal was to provoke nuclear war. Here's a for-instance:
One of LeMay's US reconnaissance crews remembered flying a B-47 deep into the USSR on May 8, 1954, and taking damage from a MiG-17. The mission made it back to England leaking fuel. LeMay ordered the crew to the US, the pilot, Hal Austin, recalled many years later:
[LeMay] said, “I tried to get you guys a Silver Star,” but he said “you gotta explain that to Congress and everybody else in Washington… so here's a couple of [Distinguished Flying Crosses] we'll give you for that mission.” There wasn't anybody in the room except the wing commander and us three guys, General LeMay and his intelligence officer…
Then General LeMay said, “Well, maybe if we do this overflight right, we can get World War III started.”
With that in mind, I'd encourage you to take LeMay's downplaying of the risks of the Cuban Missile Crisis with a grain of salt.
Second, it later came out that during the crisis the Russian commanders on the ground in Cuba had full control over the missiles that were already present in Cuba (and thus within range of the U.S.). These local commanders had the ability to launch without communication from Moscow. All it would have taken was one unusually aggressive commander to spark full scale war.
So I don't think it's fair to say that the crisis was exaggerated.
For those interested, the book is available on google books and quite interesting beyond the quote mentioned above.
In light of a blogpost by /u/restricteddata I read the other day about the development of the nuclear triad, where he touched on the mentality of superiority ingrained into the American bomber program, I am wondering how much hubris is on display here? The whole interview seems to be a largely self-congratulatory affair.
After all, and yes I know it's hindsight, it was the Soviet Union that got the better end of the deal that resolved the crisis: A guarantee of non-intervention in Cuba and withdrawal of nuclear missiles from Italy and Turkey.
It would be interesting to hear about the Soviet leadership and their rationale during the crisis.