The second prong of the Pedantic Protestant's taxonomy of Catholic apologetics is what could probably be called the evidential argument for Catholicism. The essence of the argument is an attempt to convince the Protestant of the veracity of a Catholic distinctive based on extra-Scriptural evidence. Where the PP's argument was particularly astute was in his explanation of why he finds Scriptural evidence persuasive, and that's an important element of the argument that one rarely sees given any attention. The reason I mention it is that this is one of the major distinctions between Catholic and Protestant thinking on the subject.
Catholics have difficulty expressing the concept, but the intuitive feeling on the part of most Catholics is that it is inconsistent for Protestants to accept Scriptural evidence but reject extra-Scriptural evidence. They feel (probably quite rightly) that the same set of evidence that they see as supporting Catholic claims is essential to the case for Scripture's infallibility. As St. Augustine puts it:
But if haply you should succeed in finding in the gospel an incontrovertible testimony to the apostleship of Manichaeus, you will weaken my regard for the authority of the Catholics who bid me not to believe you; and the effect of that will be, that I shall no longer be able to believe the gospel either, for it was through the Catholics that I got my faith in it; and so, whatever you bring from the gospel will no longer have any weight with me.
Now that might be a bit hyperbolic, but the point is essentially valid. For a Catholic, an argument from Scripture that defeats Catholicism is generally self-defeating, or at least, it entails a reevaluation of the entire Christian religion, not simply the Catholic distinctives. The least convincing claim possible from a Catholic perspective is the assertion that reference to the very evidence the Catholic used to justify the infallibility of Scripture is somehow forbidden when making arguments from Scripture. Instinctively, the response is "well, why do you believe Scripture in the first place?"
The PP has done me the favor of laying out a set of reasons that I can use as an example to highlight the differences. I recognize that this is probably not comprehensive given the author's own concessions to time and length restraints (and undoubtedly, my response will be inadequate in that regard as well), but I think it may be useful nonetheless. To recap, the reasons he gave for the reliability of the NT were as follows:
(i) There is good evidence [liberal scholarship notwithstanding] that the traditional authorship claims of the NT writings are in fact correct.
(ii) The Apostles and Paul were directly commissioned by Jesus and their words were supported by miracles.
(iii) Again contradicting the views of liberal scholarship, I find the NT writings, while having a few things that are hard to rectify either exegetically or historically, have an internal doctrinal unity.
(iii) is more or less the "no-brainer" test of rationality, so I won't comment on this other than to say that if you think someone is violating it, you had better be right. (i) and (ii) are the tough ones, because I can't perceive any argument that apostolicity in and of itself confers infallibility. Apostles made mistakes; that's what one would assume even if we didn't have documentation that they did. Thus, I can't see where that argument demonstrates the degree of reliability that the PP wants it to show. I have my own theories on what would, of course, but the point is that the simple fact of having an accurate record of apostolic words (which is what (i) and (ii) show) doesn't clinch infallibility for me absent evidence that this particular piece of writing is infallible, and I suspect that the unarticulated thought behind a lot of Catholic arguments is much the same. When a Protestant says "we all agree that X is infallible," it's misleading, because the reasons for that determination are drastically different between the camps, and a conclusion drawn from infallible document X could contradict the basis that someone has for the infallibility of X.
Obviously, that doesn't cover the whole issue in depth, but maybe that will help to explain what Catholics mean when they make somewhat imprecise arguments about the need for an "infallible canon." What they are really saying is that they consider something else to be an evidential requirement for demonstrating infallibility, which doesn't strike me as a ridiculous position to take.