Paging through my beloved Languages of Australia, I came across Dixon's mention of George Grey's "second great breakthrough in Australian linguistic studies", first published in Grey's 1841 Journals of Two Expeditions of Discovery in North-West and Western Australia, During the Years 1837, 1838, and 1839. As Dixon puts it:

Grey showed not only that there was a typological similarity between the languages of Australia, he also gathered enough cognates to suggest an historical connection between them: in effect, he suggested that the languages all belonged to a single genetic 'language family'. (p11)

In other words, that section of Grey's Journals represents the "no philologer could examine them all three, without believing &c." of Australian linguistics — and, since it's 2014, both volumes of the Journals are online and we can read the whole thing for ourself. It's in Volume II, under "RADICAL UNITY OF THE AUSTRALIAN LANGUAGE THROUGHOUT THE CONTINENT."

It has hitherto been very generally believed that the languages spoken in different portions of the continent of Australia are radically distinct; and as such a circumstance, were it really the case, would tend to prove that its inhabitants originated from several separate races, it becomes rather an important matter to set this question at rest, and to endeavour to show from what cause so erroneous an opinion originated.

The arguments which prove that all the Australian dialects have a common root are:

1. A general similarity of sound and structure of words in the different portions of Australia, as far as yet ascertained

2. The recurrence of the same word with the same signification, to be traced, in many instances, round the entire continent, but undergoing, of course, in so vast an extent of country, various modifications;

3. The same names of natives occurring frequently at totally opposite portions of the continent. Now, in all parts of it which are known to Europeans, it is ascertained that the natives name their children from any remarkable circumstance which may occur soon after their birth; such being the case, an accordance of the names of natives is a proof of a similarity of dialect.

That last argument is the shakiest, especially given that in chapter 11 Grey confuses part of a kinship/class system with the concept of a "family name," but the first two are supported by a couple of nice cognate tables, including one clearly showing what would later be viewed as Pama-Nyungan pronouns: "Nganya", cf proto-PN *ngay- (first person singular nominative); "Ngalee", cf proto-PN *ngali (first person dual), etc.

Under "CAUSES OF ERROR IN FORMER ENQUIRERS," Grey also bluntly recognizes some problems with the existing data:

Up to the present time we have had only very meagre vocabularies, collected by passing strangers, each of whom adopted his own system of orthography, and the comparisons formed from such compilations must necessarily have been erroneous in the highest degree. Moreover in many instances these strangers were grossly imposed upon. One gentleman published a vocabulary of the King George's Sound dialect which has been largely quoted from by other writers; in this the numerals as high as ten are given, although the natives only count to four; and the translations of some words which he has put down as numbers are very humorous, such as: What do you mean? Get out, etc.

And here's Grey's "no philologer" summary (also quoted in Dixon, naturally):

Having thus traced the entire of the coastline of the continent of Australia, it appears that a language the same in root is spoken throughout this vast extent of country; and from the general agreement in this as well as in personal appearance, rites, and ceremonies, we may fairly infer a community of origin for the aborigines.

This talk of "tracing the coastline" actually foreshadows the main contemporary amendment to Grey's thesis: it is precisely the north of Australia, the part of the coastline that Grey does not trace, that is home to the non-Pama-Nyungan families.

(Incidentally, in Dixon's account the first "great breakthrough in Australian linguistic studies" was Governor Arthur Phillip's realization in 1791 that Australia's indigenous population did not all speak the same language, ironically enough: "I now think it very probable that several languages may be common on different parts of the coast, or inland [...]" (quoted in Dixon, p10). Thesis, antithesis, synthesis!)