Good question with concise phrasing! To paraphrase, "If faith is God's gift, then those who are not given this gift are not at fault and should not be punished (with Hell)."
The core of your question deals with the way that man's cooperation is involved in his salvation. It is an age-old question that probably reached it's peak in the Congregatio de Auxiliis dispute between the Dominicans and the Jesuits which took place in the 16th century. It is secondarily addressed in the debates on predestination and free will that have taken place between various denominations and Churches (e.g. Calvin and Pighius).
Thomas Aquinas on Divine Concurrentism
Since you quoted Thomas Aquinas let's make use of his writings to try to get at the Catholic Church's high-level theological answer.
From the part you already referenced:
As regards the second, viz. man's assent to the things which are of faith, we may observe a twofold cause... ...Therefore faith, as regards the assent which is the chief act of faith, is from God moving man inwardly by grace. (ST II II, Q6, A1)
Notice that Thomas ultimately attributes man's assent to God's grace. That is to say that man assents because God moves him by grace. Strange, no? Although we tend to think of man and God as two independent agents who compete with each other for space and power, Thomas explicitly rejects this understanding of God. Let's look at what he says in his writings on free will.
First let's look at Thomas' summary of the basic natural counterargument to the above:
Objection 3. Further, what is "free is cause of itself," as the Philosopher says (Metaph. i, 2). Therefore what is moved by another is not free. But God moves the will, for it is written (Proverbs 21:1): "The heart of the king is in the hand of the Lord; whithersoever He will He shall turn it" and (Philippians 2:13): "It is God Who worketh in you both to will and to accomplish." Therefore man has not free-will.
How does he answer this objection?
Reply to Objection 3. Free-will is the cause of its own movement, because by his free-will man moves himself to act. But it does not of necessity belong to liberty that what is free should be the first cause of itself, as neither for one thing to be cause of another need it be the first cause. God, therefore, is the first cause, Who moves causes both natural and voluntary. And just as by moving natural causes He does not prevent their acts being natural, so by moving voluntary causes He does not deprive their actions of being voluntary: but rather is He the cause of this very thing in them; for He operates in each thing according to its own nature. (ST I, Q83, A1)
Thomas says the same thing in other places, such as when he talks about grace. Basically Thomas disagrees as soon as someone says, "If God did it, then I didn't do it. And if I did it, then God didn't do it." Thomas usually speaks about this in terms of first and second causes. This means that God is a kind of transcendent first cause, altogether different from any reality we might encounter in the creaturely realm. God moves everything according to its nature, whether it be rocks, plants, animals, humans, angels, or anything else which exists. In him we "live and move and have our being" (Acts 17:28). Everything we have is from God, including our free will. God causes the existence and exercise of our free will (See: Divine Conservation vs Divine Concurrentism). Nothing at all happens without God, not even the use of faculties which he has bestowed. Yet this does not mean that our will is coerced or determined (Quaestiones Disputatae de Malo, Q6, A1). The Ground of Being is not in competition with created beings.
What does this mean? In short it means that God's causal activity does not preclude free will. If we follow Thomas and say that God gives the gift of faith and the grace that causes man to assent, this does not mean that man's free will is not involved in the assent. The assent comes from man. It also comes from God. To use the scholastic language, it comes from God as first cause and man as second cause. There is a mystery here, but the mystery centers on the nature of God as transcendent cause. What's important with respect to your question is that the atheist's free will is still at work.
Dumbing it down: Sufficient Grace
Let's take a moment to move away from Thomas down to a simpler level of analysis, to the (sometimes dreaded) concept of sufficient grace. Basically the Catholic Church says that God gives all men sufficient grace to have faith and be saved. Off the top of my head, two times this issue came up was around the 4th-6th centuries (Augustine to the Second Council of Orange), and the Jansenism of the 17th and 18th centuries. Since I don't have any resources with me other than the internet, let's just look at the first three Papal condemnations of Jansenism (many of which also relate to Calvinism):
- that there are some commands of God which just persons cannot keep, no matter how hard they wish and strive, and they are not given
the grace to enable them to keep these commands;
- that it is impossible for fallen persons to resist sovereign grace;
- that it is possible for human beings who lack free will to merit; (Jansenism - Wikipedia)
(Remember that these are condemned propositions that are attributed to the Jansenists. According to the Catholic Church, they are three false propositions. They are not the Catholic Church's positive teaching, they are something She condemns.)
The most relevant to your atheist is #1. God doesn't command the impossible; he always gives the grace necessary to fulfill his commands. In Catholic tradition this grace is referred to as sufficient grace. That is to say, he gives sufficient grace for the atheist to assent and have faith. If the atheist does not have faith, it is not because God did not give him sufficient grace to do so. God always gives sufficient grace.
(Remember that this is a different level of analysis than that of Thomas Aquinas above. Here we are not probing into the mysterious nature of God's causality and trying to understand the exact way in which he relates to man's free will. Here we are just looking at Catholic doctrine on the question, which is not going to go into the subtle details that Thomas [and his successors] address.)
The Kindergarten Answer
Let's drop down one more level. Children in Catholic schools would be taught that God offers everyone the gift of faith and each person gets to decide whether or not to accept that gift. Professional theologians might cringe at this explanation, but it's arguably not as far from the truth as some like to claim. Since you quoted Thomas on a difficult matter, I tried to give the high-level explanation, but for most people the kindergarten answer would suffice.
(According to Catholicism it might be said that theologians like Jansen and Calvin ought to have just stopped at this elementary level instead of moving on to the theological calculus, becoming confused, and erring.)