St. Thomas states in many places that every agent acts for an end. At times he appears to take this as evident, but he also argues for it directly:
I answer that, Every agent, of necessity, acts for an end. For if, in a number of causes ordained to one another, the first be removed, the others must, of necessity, be removed also. Now the first of all causes is the final cause. The reason of which is that matter does not receive form, save in so far as it is moved by an agent; for nothing reduces itself from potentiality to act. But an agent does not move except out of intention for an end. For if the agent were not determinate to some particular effect, it would not do one thing rather than another: consequently in order that it produce a determinate effect, it must, of necessity, be determined to some certain one, which has the nature of an end. And just as this determination is effected, in the rational nature, by the “rational appetite,” which is called the will; so, in other things, it is caused by their natural inclination, which is called the “natural appetite.”
Basically his argument is that an agent is doing something, and there must be some explanation for why it is doing what it is doing, rather than something else. And a final cause is nothing but such an explanation.
Now someone might object that a final cause is this sense is more general than acting for an end, and certainly more general than desiring an end. For example, logical necessity may be a final cause in this sense. Thus if we ask why I walk, rather than both walking and not walking at the same time and in the same way, the logical impossibility of the latter is a sufficient explanation. Or again, if we ask why a very intelligent person does not win at Tic-tac-toe against a relatively unintelligent one, but instead ties, the fact that there are strategies in the game that cannot be defeated, and that are well known even to unintelligent persons, is a sufficient explanation. Far from implying desire, such explanations may be contrary to desire: the person may desire to win the game, but cannot do so.
According to this objection, the fact that a rock falls may have some explanation, but there is no reason to think that the explanation would be that it desires to fall or to be at the center, or that it has any kind of desires at all.
The answer to this is that we must distinguish between what is material in desire, and what is formal. The fact that desires are something that we feel is material in them, and is not why we call them desires. As noted in the linked post, it is not from the sensible experience that we know our desires are desires rather than some other kind of feeling, but from the fact that when we have them, we tend to do certain things. Thus, the feeling is material in desire, while the tendency to do something is formal. Now in the case of the rock, there are no strong reasons for supposing that they have any feelings, and thus for supposing that they have what is material in desire. But it is evident that they have a tendency to do something, and this is what is formal in desire, and constitutes the real reason for saying that something is a desire rather than something else.
It is correct, then, to say that St. Thomas’s universal claim is an analogous extension of the ideas of desire and of intending an end. Nonetheless, it is a perfectly reasonable one and conforms precisely with the formal meaning of these terms.