Claims like "many philosophical works have been vitiated" by blurring the use/mention distinction are, shall we say, greatly exaggerated. They sensationalize even Leśniewski and Quine, the main champions of the distinction in its heyday, who were less sweeping. Leśniewski initially came to distrust symbolic methods generally after detecting use/mention issues in Russell and Whitehead's Principia Mathematica, but he got over it after reading Frege, and Quine saw it as far less consequential.
One can see Tarski's distinction between object and meta language as an outgrowth of the use/mention (especially in Russell-Whitehead's context), and it did, arguably, play a role in the early misunderstanding of Gödel's results by Russell, Wittgenstein and Zermelo, see What sources discuss Russell's response to Gödel's incompleteness theorems? That would be as glaring a "vitiation" (of their criticisms) as I can think of, but I doubt a 16 year old would appreciate it. See also What are some non-trivial examples of the use/mention error? for Azzouni's recent example of misunderstanding rigor of mathematical proofs in the literature due to use/mention and object/meta confusions.
But, like its progenitor, Tarski's distinction came to be seen as artificial as applied to the natural language. Quine's influential student, Davidson, argued that use and mention are naturally mixed to begin with, and questioned the significance ascribed to the distinction altogether, see his Quotation. It remained dubious since, beyond detecting some plain beginner errors. In particular, concerning the double quotation device that Devitt and Sterelny adopt in the book, Davidson confesses:
"My plan was to use single quotation marks when I wanted to
refer to the expression a token of which was within, but double quotation marks when I wanted to use the expression in its usual meaning while at the same time indicating that the word was odd or special ("scare quotes"). I blush to admit that I struggled with this absurd and unworkable formula for a couple of years before it dawned on me that the second category contained the seeds of its own destruction."
As for Quine, in Mathematical Logic §4, the locus classicus on the matter, he restricts the discussion, at most, to logical issues in philosophy of language:
"In the literature on the logic of statements, and in other foundational
studies of mathematics as well, confusion and controversy have resulted from failure to distinguish clearly between an object and its name... But it is primarily in mathematical logic that carelessness over these distinctions is found to have its more serious effects. At the level of the logic of statements, one effect is obliteration of the distinction between predicates of statements and composition of statements — a distinction which will be considered in the next section."
In the next section the culprits identified by name are the same as those named earlier by Leśniewski, Russell and Whitehead in Principia Mathematica. But not even Leśniewski suggested that Principia was thereby "glaringly vitiated". Quine surmises quite the opposite, that the problem does not go "beyond the level of unfortunate exposition and nomenclature":
"In Whitehead and Russell's exposition and terminology the distinction between predicate and statement connective is blurred. The notation '— ⊃ —' is explained indiscriminately in the sense of the truth-functional conditional and in the sense of material implication. It is translated not only thus:
(14) If — then —
but also thus:
(15) If — is true then — is true,
(16) — is false or — is true,
(17) — implies —.
... In all technical developments the expressions which Whitehead and Russell adjoin to the sign '⊃' have the form of statements rather than names. The mere fact of its iteration indeed, e.g. in the manner '— ⊃ (— ⊃ —)', is enough to determine the sign as a statement connective rather than a predicate about statements. In short, the versions (15)-(17) do not operate in Whitehead
and Russell's work beyond the level of unfortunate exposition and nomenclature. The English idiom which '— ⊃ —' supplants in practice is not (15), (16), or (17), but (14)."