I rather suspect that most non-academics' experience of The Greek Myths centres not on the interpretations of myth, but upon the narratives of the legends. Many readers, indeed, have no choice but to experience the work in this way, as they will be reading one of the condensed versions (e.g. this), where the abridgment is achieved by the simple expedient of leaving out all the explanatory material. Taken on this level, one can see why someone like Margaret Atwood might describe The Greek Myths as 'crucial'. Because there really isn't anything else quite like it for providing a narrative summary of the mythological stories, together with variant versions, unless you go back to the works of the Edwardian A.R. Hope Moncrieff or the Victorian Thomas Bulfinch, both of whom tend to record only one 'canonical' version. Despite certain oddities of presentation, such as the fate of Agamemnon being dealt with before the chapters on the Trojan War, rather than after, as one might expect, Graves has an accessibility the earlier works lack.
For my own part, I would probably these days first consult Cassell's Dictionary of Classical Mythology by Jenny March, or the Penguin Dictionary of Classical Mythology (a translation and abridgment of Pierre Grimal's Dictionnaire de la Mythologie Grecque et Romaine) if I needed to do any mythological research. But both of those, as Lowe notes, are alphabetic reference works. I would soon turn to Graves for narrative, and all three books were within reach as I was writing my review of Gemmell's Troy.
If Graves is ever to be replaced, then it needs to be by a work that matches or improves upon what Graves does well, providing narratives of mythology with all variants, and the original sources (even if Graves lifted those from elsewhere), ideally placed into a historical context, so one knows which sources are earliest, and discards what Graves does badly (i.e. the commentary), and does all this in a mass market edition. I've not read Songs on Bronze, but from the reviews (and a brief browse in the British Museum bookshop) it is apparent that this is not that work. Rather, it is a novelistic retelling of certain stories that does not admit of variation. Nor does it appear from a brief browse of the pages available on Amazon that Timothy Gantz's Early Greek Myth is the Graves replacement classicists hope for, though the compendious collection of references therein suggests that I need a copy on my shelves.