This is almost the exact opposite of the truth. Greenspan's conduct as Fed Chair was a betrayal of his earlier expressed views, a betrayal that libertarians and Randians have complained about many, many times over the decades.
The truth has been explained very by Richard Spencer here. In the early sixties, Greenspan wrote an excellent essay, believe it or not, defending the gold standard, on the grounds that it would prevent precisely those policies that he did pursue years later, after he had become our monetary dictator.
In that essay, he criticized a fateful move on the part of the Fed in 1927, when it decided
to assist Great Britain, who had been losing gold to us because the Bank of England refused to allow interest rates to rise when market forces dictated (it was politically unpalatable). The reasoning of the authorities involved was as follows: if the Federal Reserve pumped excessive paper reserves into American banks, interest rates in the United States would fall to a level comparable with those in Great Britain; this would act to stop Britain’s gold loss and avoid the political embarrassment of having to raise interest rates.Greenspan's essay is still in print and still well worth reading.
The ‘Fed’ succeeded; it stopped the gold loss, but it nearly destroyed the economies of the world in the process. The excess credit which the Fed pumped into the economy spilled over into the stock market—triggering a fantastic speculative boom. Belatedly, Federal Reserve officials attempted to sop up the excess reserves and finally succeeded in braking the boom. But it was too late: by 1929 the speculative imbalances had become so overwhelming that the attempt precipitated a sharp retrenching and a consequent demoralizing of business confidence. As a result, the American economy collapsed.
What I find seriously spooky about it is that in it he is quite explicitly saying that policies substantially similar to the ones he later enacted eventually lead to speculative booms, crashes, and depressions. As Spencer points out, after Fed interest rates reached an all-time high of 19 % under Greenspan's predecessor, he eventually hammered them down to as low as 1 %. This is just the sort of speculation-triggering inflationism he warned about back when he was a Randian. Of course he was right the first time. These policies have indeed lead to disaster, and will continue to do so.
Moyers' comments are one more example of the often-repeated myth that these events were caused by laissez-faire caplitalism, whereas they were caused by government intervention. In fact, critics of interventionism predicted it would do so, including even the Arch Intervener himself, in a earlier lifetime.
(Notice that the people who were suprised when the catastrophe struck were proponents of intervention, like Ben Stein and Paul Krugman. We on the other side had been warning it was coming for a while now, and were in large part well prepared for it.)
The story of Allan Greenspan is a very sad one and a very old one. Once upon a time he believed in freedom and the market, and in the principle that working and saving are virtues. Then one day Ronald Reagan offered him absolute power over the US money supply, and it corrupted him. He went over to the great Keynesian transvaluation, in which working and saving are less virtuous than consuming and spending.
As Ellsworth Toohey, villain of The Fountainhead, says to one of his victims, "Ever read Faust, Peter?" Political power is indeed a deal with the Devil.
Added Later: On related themes see Harvard (!?) economist Jeffrey Miron. Also see the always-excellent George Reisman. Also check out this audio of Ron Paul commenting on the Greenspan betrayal. Finally, my friend the estimable Richard Epstein has weighed in.