Whately's discussion occurs in the Introductory Lessons on Morals. The context is actually quite important. One of the themes of this work is that human beings have a moral faculty, which we call conscience, and one of the subthemes is that all moral rules and standards are addressing this moral faculty, which they presuppose. Thus when Whately 'criticizes' the Golden Rule, he is really arguing that Scriptural moral guidance by its nature presupposes our ability to discern right and wrong. It does not follow that he rejects it or considers it to be unimportant. The Golden Rule, which is his example, cannot be the foundation of morality, but it can be a principle for cultivating and correcting our conscience. His analogy is that of a clock and a sundial, with the clock standing in for conscience and the sundial for Scriptural rules. If you want to tell time, the clock is in general the most convenient way, since it is always available and can be placed anywhere, whereas the sundial can only be used when the sun shines on it. But clocks that go wrong go consistently wrong and cannot correct themselves. If you want to make sure your clock is keeping accurate time, you need to correct it occasionally on the basis of a more reliable timepiece, like the sundial. Thus he concludes (p. 30):
We must be careful, therefore, to regulate both our business by the clock, and the clock by the dial; that is, to regulate our conduct by our Conscience, and our Conscience itself by the commands and instructions which God has given us.
The Golden Rule in particular, as Whately sees it, serves to counterbalance the bias of self-interest. Although he does not say explicitly, I think Whately here is drawing from Butler, who in his sermon on self-deception notes our excessive partiality to ourselves and uses precisely the example that Whately uses: David and Nathan. David, having committed adultery with Bathsheba and caused her husband to be slain, had less of a sense of the wrongness of the act than he did from Nathan's story of a rich man stealing a poor man's lamb; he did not apply his correct principles in the latter to himself until Nathan turned the story on him. Thus the Golden Rule serves as our Nathan:
And we, if we will make a practice of applying the golden rule, may have a kind of prophet always at hand, to remind us how, and when, to act on our principles of right.
Thus it is misleading to speak of Whately's "criticism" of the Golden Rule. He fully affirms it and insists that it is of extraordinary moral value. What he criticizes, rather, are relatively thoughtless interpretations and applications of it -- interpretations that, it should be said, he does not attribute to anyone or regard as particularly plausible interpretations of the rule in the first place. [ADDED LATER: It's perhaps worth insisting on this latter point: the whole structure of Whately's argument requires the interpretation that he thinks pretty much everyone can see at once that these absurd, wrong, or impossible applications are problematic, and this is further required by his basic thesis that such rules are directed to our innate ability to think morally.] People do not, in fact, tend to interpret the rule in the absurd, wrong, or impossible ways with which Whately opens his discussion -- they are raised in order to open a larger discussion of the relationship between conscience and the moral guidance of Scripture.