The origin to the idea that Montgomery claimed to have rescued an incompetent US command at the time of the Ardennes offensive, is almost entirely due to a cleverly produced piece of German propaganda that was put out in the guise of a BBC announcement. Designed, as it obviously had been, to drive a wedge between the two western Allies, it was clearly successful - as witnessed by the fact that writers such as the one quoted in the question are still, 78 years later, confused about the facts. 

[You can read all about it in this essay (2020) by staff of the National Archives in Britain.](https://blog.nationalarchives.gov.uk/how-nazi-fake-news-split-allied-commanders-in-1945/)

Initially all concerned - including American forces all over Europe - fell for what  appeared to be a BBC report. Unfortunately the matter was taken up by some of the more jingoistic press in Britain - notably the *Daily Mail* - for whom it wouldn't have taken its editor much to fall for such a piece of  "fake news".

From what I have read elsewhere, there may be some evidence that Montgomery warned Eisenhower, a couple of weeks before the German offensive, that it was likely to happen due to the large gap in the Ardennes front, held only by troops that were "resting". And he may already have urged Eisenhower to move Patton's forces north at that time. 

Remember the British commanders had had greater experience of battle - from WW1 - than their American colleagues had had. But I strongly suspect that Monty, having previously not withheld his contempt of some of the US's less-experienced commanders, ( especially in Italy) was not taken as seriously as he should have been on this occasion. This was surely an important lesson for both sides. 

As for the Ardennes battle, the facts of the numbers killed must leave in no doubt the question of who bore the brunt of the German offensive. American dead - circa 75,000 British dead circa - 1,500. But let's not overlook the 120,000 ordinary decent German soldiers who also lost their lives in a frantic last gamble by the maniac who governed them.