For a while, I've been meaning to comment on the "that"/"which" controversy, the claim that "which" should not be used with restrictive relative clauses, nor "that" for nonrestrictive. From a linguistic point of view, it seems clear that this view is descriptively barren. Geoff Pullum provides a convincing and entertaining argument on Language Log, based on the sentence "The key point, that all the popular reports missed, is that FOXP2 is a transcription factor...". The rarity of sentences like these, in which "that" is used for a nonrestrictive relative clause, leads Pullum to refer to it as "ivory-billed".
I suppose, and am happy to stipulate for the purposes of discussion, that the use of "which" for restrictive relative clauses and "that" for nonrestrictive (or supplemental, as Pullum prefers) is grammatical. Nonetheless, the overwhelming preponderance of occurrences of "which" for nonrestrictive clauses means that the use of "that" in that context is much more likely to give pause to the reader, a kind of cognitive setback. For that reason, a charitable writer (and shouldn't we all strive to be one of those?) ought to use "which" for nonrestrictive relative clauses -- not because it is "wrong" to use "that", or ungrammatical, but because the use of "that" is likely to be jarring to a significant fraction of one's readers. (And I don't only mean the Fowler-type prescriptivist readers, though I suppose there's no reason to be jarring them needlessly either.) An excellent point of evidence is the fact that Pullum had to ask the author directly which meaning he had intended in the ivory-billed sentence; had he used a "which", no clarification would have been needed.
In the particular case of the sentence quoted above, there is no concomitant advantage to using "that" over "which" that would compensate for the negative effect of jarring or confusing the reader. Thus, its use should be prescriptively deprecated. (This issue of compensation allows me to avoid proscriptions against splitting infinitives or dangling prepositions, the slavish following of which leads to circumlocutions and semantic errors. Avoiding these negative effects clearly compensates for the oh so very slight jarring effect on some small fraction of true-believing Fowlerians.) By a similar argument, the use of "which" for restrictive relatives should be deprecated as well in formal writing.
What I am arguing is that even though the language does not enforce the distinction between nonrestrictive and restrictive in terms of "which" versus "that" (and commas versus none), respectively, there is still a good reason to write as if it did. There was nothing wrong in the quoted sentence even under the intended interpretation, just something infelicitous.
Am I trying to have my cake and eat it too? To be able to rail prescriptively while keeping my linguistic descriptivist moral stance? Yes.