I think part of the confusion here is an ambiguity in "an effect cannot be greater than its cause". A better statement of this principle is (the central Thomist principle that) "one cannot give that which one does not have". But one can "have" things in different ways. A being could possess an ability to speak infallibility "naturally" (as the Holy Spirit does) or it could possess it "unnaturally" (that is, it could be endowed with this ability by something outside itself, like Balaam's ass).
PCA Confession Protestants (which I reference b/c that's my background, but also b/c I think its pretty representative of conservative Protestants as a whole) believe that the Church (qua Church as opposed to the individual authors of the NT) never possessed infallibility either naturally or unnaturally as I have defined those terms. And yet PCA Confession Protestants agree that some interpretations are inspired (e.g. the doctrines of the Trinity and of Christ's nature, both of which are enunciated in the PCA’s confession), while others (e.g. unitarianism and Arianism) are heretical. On what basis is this distinction made?
Biblical interpretation? This won't work. The Bible admits of many interpretations, none of which is self-authenticating. Who is to say that the contrary interpretation of such doctrines is heretical? The Bible cannot resolve such debates because the Bible’s ambiguity is the source of the controversy. To say that all contrary interpretations are “unreasonable” is only to beg the question.
The only unambiguous evidence of the truth of these doctrines is that the Church pronounced that them as such. Thus it seems that in order to confidently say that these doctrines are true, PCA Confession Protestants must also say that they were infallibly pronounced. This is a difficult distinction (i.e. between truth and infallible pronouncement), so here's an analogy: Jewish historian Pinchas Lapide believes that the doctrine of the resurrection of Jesus is true (that it actually happened), but he does not believe that Paul's pronouncement of it in 1 Corinthians was infallible. The problem for PCA Confession Protestants is that (unlike the resurrection) there is not enough independent evidence of the truth of doctrines like the Trinity, Christ's dual nature, or the non-canonicity of the Gospel of Peter to think of them as dogma rather than mere speculative opinion. Speculative opinion is ok for thinks like the Shroud of Turin, but, as Mathison says, it destroys Christianity when it expands to doctrines as central as the Trinity, etc.
So, PCA Confession Protestants must believe that certain doctrines were infallibly pronounced (or else treat them, not as doctrine, but as mere speculative opinion and destroy Christianity). But, since "one cannot give that which one does not have":
(1) If a doctrine is infallible, then the body that pronounced the doctrine must have possessed the ability to act infallibly as it pronounced the doctrine.
And, as I have argued above:
(2) The PCA Confession treats doctrines such as the Trinity and Christ's nature as infallible (not just true)
(3) Therefore, doctrines such as the Trinity and Christ's nature must have been pronounced by a body that possessed the ability to act infallibly
(4) But the PCA says that no body that ever pronounced these doctrines possessed the ability to act infallibly.
(5) Therefore, the PCA confession suffers from a logical contradiction that can only be resolved by Church infallibility.
Ironically, this is the same style of argument that the great PCA pastor Tim Keller uses with great effect to expose the contradiction faced by moral relativists.
Keller argues that moral relativists like Anthropologist Carolyn Fluehr-Lobban (and, closer to my heart, Yale Law Professor Arthur Leff) have noted that their major premise of moral relativism entails the conclusion that they must not export western moral values (against honor killing, female circumcision, etc.) upon cultures that do not recognize western values. But they know that this conclusion is wrong. They know that honor killing and female circumcision is wrong and should be stopped whether these cultures agree or not. In response, they just throw up their hands and just deal with the cognitive dissonance. But Keller argues that if a major premise leads invariably to a conclusion you know is wrong, then you should revisit the major premise.
Similarly, PCA Protestants know that the doctrines of the Trinity and so forth are not mere speculative opinons, but divinely revealed infallible truth. But their major premise that the Catholic Church has never been endowed with infallibility leads them to the conclusion that these are mere speculative opinions. So, if they are to be consistent, they should revisit their major premise. Maybe the Catholic Church has been endowed with infallibly.