Premise: I do not have Buxton's book, so my objections are based on other sources. The origin of this claim are to be traced in a series of references. These include:
Children of inferior parents, and of the better, when they chance to be deformed, will be put away in some mysterious, unknown place, as they should be.
Plato, The Republic, 461 C
As to exposing or rearing the children born, let there be a law that no deformed child shall be reared;
Aristotle, Politics, 1335b 19-20
Other sources include Aristophanes and Euripides. All of these sources are of accessory character. The fact that philosophers advocate a practice can mean that such a practice was not abhorred, but does not mean that it was common use. As for theater, and especially Euripides, one must consider that art often deals with exceptional situations, e.g. an exposed baby who reverses his faith and becomes king of Thebes.
An authoritative source is instead Polybius, XXXVI, 17
For as men had fallen into such a state of pretentiousness, avarice, and indolence that they did not wish to marry, or if they married to rear the children born to them, or at most as a rule but one or two of them [...] For any ordinary man will tell you that the most effectual cure had to be men's own action, in either striving after other objects, or if not, in passing laws making it compulsory to rear children.
(emphasis added). This refers only to the Hellenistic period, as it's clear from context (ibid.):
In our own time the whole of Greece has been subject to a low birth-rate and a general decrease of the population, owing to which cities have become deserted and the land has ceased to yield fruit [...]
So to sum it up, the practice was almost certainly in use in Athens, as it was in many other cultures at the time. There is however no clear evidence that it was common use, certainly nowhere near its practice in Sparta, where it seemed universal. We have suggestions that the practice might have been common in the Hellenistic period, in the whole of Greece.
The Exposure of Infants in Athens, by La Rue van Hook, contains a list of authors who support instead the view reported in Wikipedia.
La Rue van Hook, Transactions and Proceedings of the American Philological Association, 1920, 51, 134-145