The charge of a quantifier shift occurs at a part in the argument that is trying to prove the existence of something like matter or natural laws, which is why (a) there is no quantifier shift and (b) no one would care if there were.
(a) The charge is that it is fallacious to move from saying that if everything did not exist at some time then there is some time at which everything did not exist. First off, STA only makes this claim about the generated, not about everything – his argument is in fact trying to show that the generated be less universal than everything.
Take all generated things, which in STA’s sense means things that did not exist at some time and later did. Maybe this is an infinite set (like prime numbers) or maybe it isn’t. Maybe it’s existed for all time, and maybe it hasn’t. The question is whether everything could be this sort of thing. STA says no since if this would require that natural generation would reduce to nothing at all, and so could not even occur naturally.* By “the natural” he’s including things like matter, absolute space and particles, mass-energy, quantum states, natural laws, and/ or whatever background stuff our physical theories might reduce a universe or multiverse to. In other words, he’s trying to give a reason why every physical theory from Thales to Sean Carroll says that the fundamental physical reality is something everlasting and ungenerated. STA isn’t saying that if the set of generated individuals must have a some time at which they don’t exist, but that if there is no such thing as matter, absolute space and particles… etc. then generated things could not arise naturally, though they obviously do.
(b) Most persons would simply grant STA’s point without argument, or by an inductive argument that pointed to every physical theory ever devised, which seeks to reduce the generated (in STA’s sense) to the ungenerated. All the evidence we have of seeking to understand nature, whether by science, art or myth, points to the axiomatic character of reducing the finitely-temporal and compound to the everlasting and simple.
There are objectors, of course. Smolin and Unger want to deny the axiom, and this would put them in a tradition that seems to begin with Nietzsche who bases his whole philosophy on the mistake of trying to reduce the phenomenal and evanescent world to a deeper unchangeable reality (see “Reason in Philosophy” in Twilight of the Idols for his simplest and most magisterial treatment).** While Heidegger’s thought is incomplete and with at least one fundamental shift, he seems to want to articulate some sort of fundamental reality to the purely temporal. At minimum, he clearly sees this Nietzschean pan-temporality as the fundamental problem of contemporary thought, and in this he sees a good deal further and more clearly than almost everyone.
This helps to manifest just how far the charge of the quantifier shift misses the mark. The argument STA actually gives is a refutation of physical nihilism and a defense of physics as a fundamental science. Cosmological arguments usually articulate diverse levels of reality that are more and less godlike (a point that will be made explicitly in the Fourth Way) and the Third Way does this in a way that establishes the godlike character of the simple and everlasting entities that we seek in order to articulate a physical theory.
*If something natural could not occur naturally, it either occurs from some other-than-natural means or it doesn’t occur at all, and STA takes the second fork in his argument. There’s more than one reason to do this (1) it allows him to prove the existence of a deus in the hardest possible case and (2) it establishes a level of godlike existence in nature itself, which both gives us an analogy to understand God and makes for a greater manifestation of his power.
**The doctrine of “eternal return” is not an exception to this, but is either a clumsy and ill-fitted addition to his thought or a metaphor for the purely temporal existence of all things.