Did Malcolm Fraser save NATO?

In an interview with Mark Colvin, former Prime Minister Malcolm Fraser recounts a conversation with Vice-President Bush in 1982, during which Fraser is informed that President Reagan's UN Ambassador, Jeanne Kirkpatrick, is pushing Reagan to support Argentina after Argentina had invaded the Falkland Islands, a UK territory in the South Atlantic. Fraser doesn't think this is such a good idea:

“Well, have you thought how your most important NATO ally in Europe will react? If you support Argentina against Britain, do you think Margaret Thatcher will just sit down and take it? Or do you think she’ll condemn the United States for being an unreliable ally from one end of Europe to the next, and if she ever gets to Washington again, she’ll do it from the heart of Washington also. And there are many people around the world who would think that she has great justice on her side. And what will be left ,what will be left of NATO; will you still have NATO at the end of that argument?”

According to Fraser, Bush then abruptly cuts short their dinner to make a call to Washington, where Kirkpatrick is putting her case to a National Security Council meeting. He returns 90 minutes later:

...he came out about an hour and a half later, thumbs up: “It’s all right Malcolm, we’re supporting Margaret. And if you hadn’t keyed yourself into that meeting, Jeane would have won that argument in ten minutes”’.

If there are any readers out there with a deep knowledge of this period, I'd love to hear from them, because although Fraser's account may be accurate, it seems a trifle self-serving. Is it likely that the Vice-President hadn't thought through the implications of backing Argentina before Fraser intervened, or that no-one else had raised these issues with Reagan? As Colvin says in his piece, the Reagan Administration was divided on what to do about the Falklands, suggesting at least some of Reagan's senior advisers wanted to back Thatcher.

So was Fraser's intervention really so decisive?