He quotes two articles: II-II Q4 A8 ("Whether faith is more certain than science and the other intellectual virtues?") and II-II Q8 A5 ("Whether the gift of understanding is found also in those who have not sanctifying grace?"). Aquinas' answer to the first is "Yes, it is," and to the second he replies that they do not.
The problem, as I see it, is that there was a particularly apologetic bent to the discussion in which TF had previously participated, and the passages from the Summa that he quotes are not directed towards the question of apologetics. He summarizes this new post this way:
[T]ools like the Transcendental Argument for the existence of God (TAG) can be a useful negative took to expose presuppositions, but it cannot be a complete defense of the faith - it cannot be a stand-alone apologetic. Human reason must be subordinated to divine truth, even though human reason is an instrument and tool by which and through which we understand.Well, the problem is that the average unbeliever can hardly be expected to agree that he must subordinate his reason to the Bible. And St. Thomas recognizes this.
For example, when discussing whether the Holy Trinity can be known by natural reason, St. Thomas answers in the negative:
It is impossible to attain to the knowledge of the Trinity by natural reason. For, as above explained (12, 4, 12), man cannot obtain the knowledge of God by natural reason except from creatures. Now creatures lead us to the knowledge of God, as effects do to their cause. Accordingly, by natural reason we can know of God that only which of necessity belongs to Him as the principle of things, and we have cited this fundamental principle in treating of God as above (Question 12, Article 12). Now, the creative power of God is common to the whole Trinity; and hence it belongs to the unity of the essence, and not to the distinction of the persons. Therefore, by natural reason we can know what belongs to the unity of the essence, but not what belongs to the distinction of the persons. [I Q32 A1]This is presumably non-controversial - at least to some extent. But St. Thomas continues:
Whoever, then, tries to prove the trinity of persons by natural reason, derogates from faith in two ways.More importantly in the present context, though, is the second way:
Firstly, as regards the dignity of faith itself, which consists in its being concerned with invisible things, that exceed human reason; wherefore the Apostle says that "faith is of things that appear not" (Hebrews 11:1), and the same Apostle says also, "We speak wisdom among the perfect, but not the wisdom of this world, nor of the princes of this world; but we speak the wisdom of God in a mystery which is hidden" (1 Corinthians 2:6-7).
Secondly, as regards the utility of drawing others to the faith. For when anyone in the endeavor to prove the faith brings forward reasons which are not cogent, he falls under the ridicule of the unbelievers: since they suppose that we stand upon such reasons, and that we believe on such grounds.Now unbelievers do not receive the authority of Scripture; hence, St. Thomas says, one opens himself to ridicule if he appeals to the Bible in defense of that which can only be known by faith when attempting to persuade unbelievers concerning the Faith. And this is not the only place in the Summa Theologica where he makes this argument; he says much the same thing, for example, in I Q46 A2 in discussing the question of whether the creation of the world can be proven by reason or must be accepted as an article of faith:
Therefore, we must not attempt to prove what is of faith, except by authority alone, to those who receive the authority; while as regards others it suffices to prove that what faith teaches is not impossible. Hence it is said by Dionysius (Div. Nom. ii): "Whoever wholly resists the word, is far off from our philosophy; whereas if he regards the truth of the word"--i.e. "the sacred word, we too follow this rule." [emphasis added]
By faith alone do we hold, and by no demonstration can it be proved, that the world did not always exist, as was said above of the mystery of the Trinity (32, 1). The reason of this is that the newness of the world cannot be demonstrated on the part of the world itself. For the principle of demonstration is the essence of a thing. Now everything according to its species is abstracted from "here" and "now"; whence it is said that universals are everywhere and always. Hence it cannot be demonstrated that man, or heaven, or a stone were not always. Likewise neither can it be demonstrated on the part of the efficient cause, which acts by will. For the will of God cannot be investigated by reason, except as regards those things which God must will of necessity; and what He wills about creatures is not among these, as was said above (Question 19, Article 3). But the divine will can be manifested by revelation, on which faith rests. Hence that the world began to exist is an object of faith, but not of demonstration or science. And it is useful to consider this, lest anyone, presuming to demonstrate what is of faith, should bring forward reasons that are not cogent, so as to give occasion to unbelievers to laugh, thinking that on such grounds we believe things that are of faith. [emphasis added]And again in II-II, Q1 A5 ad 1:
Unbelievers are in ignorance of things that are of faith, for neither do they see or know them in themselves, nor do they know them to be credible.But if they do not consider the Bible to be credible as the Word of God, then it is pointless to attempt to persuade them on the basis of its authority.
But all of this is mere window dressing compared to the major argument: that is, the Summa Contra Gentiles! The entire work is intended as an argument for the Faith not from Scripture, but rather on the grounds of reason:
 And so, in the name of the divine Mercy, I have the confidence to embark upon the work of a wise man, even though this may surpass my powers, and I have set myself the task of making known, as far as my limited powers will allow, the truth that the Catholic faith professes, and of setting aside the errors that are opposed to it. To use the words of Hilary: “I am aware that I owe this to God as the chief duty of my life, that my every word and sense may speak of Him” [De Trinitate I, 37].So we see that St. Thomas acknowledges the weakness of reason for some things, but he insists that with unbelievers there is no other course that can be taken. Far from making his appeal from Scripture to the unbeliever, he argues on grounds that the unbeliever can accept. Hence we see that TF hasn't told the whole story about how Aquinas views the utility of Scripture for persuading the non-Christian.
 To proceed against individual errors, however, is a difficult business, and this for two reasons. In the first place, it is difficult because the sacrilegious remarks of individual men who have erred are not so well known to us so that we may use what they say as the basis of proceeding to a refutation of their errors. This is, indeed, the method that the ancient Doctors of the Church used in the refutation of the errors of the Gentiles. For they could know the positions taken by the Gentiles since they themselves had been Gentiles, or at least had lived among the Gentiles and had been instructed in their teaching. In the second place, it is difficult because some of them, such as the Mohammedans and the pagans, do not agree with us in accepting the authority of any Scripture, by which they may be convinced of their error. Thus, against the Jews we are able to argue by means of the Old Testament, while against heretics we are able to argue by means of the New Testament. But the Muslims and the pagans accept neither the one nor the other. We must, therefore, have recourse to the natural reason, to which all men are forced to give their assent. However, it is true, in divine matters the natural reason has its failings. [I, 2, 2-3; emphasis added. Note that I quote from the Pegis translation, which (as far as I know) is not legally available online; the link here is to another translation]
Update: I realize that TF concedes:
It is not that there is no reason to believe, but that some things cannot be reached by bare reason.Again, as I said at the outset, the particulars of his view aren't what concerns me here. When it comes to Aquinas' view of the role of reason in apologetics, TF's post doesn't give the whole picture. Appeals to authority are fine when we speak to those who accept the authority, but they're meaningless to those who don't.