Were the Finches important to Darwin in formulating his theory?

School children are often taught that the finches of the Galapagos Islands were very important in helping Darwin to come up with his theory of evolution. For example, the BBC GCSE Bitesize Revision Biology: Variation and inheritance states:

Whilst studying wildlife on the Galapagos Islands [Darwin] noticed that the Galapagos finches showed wide variations - eg in beak shape and size - from island to island. Darwin deduced that these differences made the finches better adapted to take advantage of the food in their particular local environment - thin, sharp beaks prevailing where the birds' main food was insects and grubs, and large claw-shaped beaks where their diet was buds, fruit and nuts. In each locality the finch population had somehow developed beaks which were suitable for that particular environment.

Darwin concluded that in each locality one or more individual finch happened to acquire, by random mutation, a beak shape more suitable for the food sources in that locality. These individuals then had a competitive advantage over their fellow finches, enabling them to grow and reproduce more successfully, and pass on their more specialised beaks to successive generations - until eventually the characteristic had spread throughout the finch population in that locality.

The BBC is mistaken in much of what it says here. When he was on the Galapagos Islands, Darwin did not notice that different islands had different finches. Neither did he realise that the finches were closely related despite their differences in beak shape. He did not match different beak shapes to different diets. Even after his return to London, Darwin's biographers note that he "remained confused by the Galapagos finches...unaware of the importance of their different beaks...He had no sense of a single, closely related group becoming specialized and adapted to different environmental niches." (p. 209, Darwin - A. Desmond and J. Moore).

Darwin did not mention the finches in his book The Origin of Species. They only appear in his Journal, being mentioned only in passing in the first edition (1839), and then having a few paragraphs and a picture six years later in the revised edition (1845). The most that Darwin would ever say about finch evolution is found here:

Seeing this gradation and diversity of structure in one small, intimately related group of birds, one might really fancy that from an original paucity of birds in this archipelago, one species had been taken and modified for different ends... Unfortunately most of the specimens of the finch tribe were mingled together; but I have strong reasons to suspect that some of the species of the sub-group Geospiza are confined to separate islands. If the different islands have their representatives of Geospiza, it may help to explain the singularly large number of the species of this sub-group in this one small archipelago, and as a probable consequence of their numbers, the perfectly graduated series in the size of their beaks. (pp403-420)

Darwin, Journal of Researches into the Natural History and Geology of the countries visited during the voyage round the world of H.M.S. Beagle, revised edition, Henry Colburn 1845.

So all Darwin did was speculate that the different finches had descended from a common ancestor and had changed to be able to do different things. He was never sure that the different species were from different islands. He certainly never came up with the detailed theory for how the finches diversified which the BBC suggests.

The BBC makes this mistake because a myth has arisen around these Galapagos Finches. They were never known as "Darwin's Finches" until 1936, and the name was popularised by ornithologist David Lack in his book Darwin's Finches (1947). Lack described the detailed account of Finch evolution, recounted by the BBC, and also promoted the myth that the finches had given Darwin important insights into evolution.