Why not say that a universal is nothing other than a collection of its species, or the species is nothing other than collection of individuals? St. Thomas never speaks to this question directly, but a fitting response can be gathered from what he says about how species arise in the process of our knowing.
Our intellect moves from what is more general to what is more distinct by adding to our general ideas, and the concept of “species” arises as the result of adding to some general idea. Species arises precisely by adding a difference to a genus. If one simply reduces the genus to a multitude of species, he annihilates the very thing that he added the difference to in order to form the species in the first place. An analogous argument applies to the distinction between the species and the individuals. There is something that makes this individual different from that one. Whatever this difference is, if we do away with the species we annihilate the very thing that allowed for opposing differences between the individuals. And so just as the very notion of species vanishes when we identify the genus with all of its species, so too the very notion of an individual vanishes when we identify the species with all of its individual members.
One might object that individuals are still different from each other even if they do not have a common genus. For example, there is no common genus of “shoe” and “white”, but they are certainly different. But this uses a notion of “different” which is not being discussed here. This kind of difference is what St. Thomas calls “other”, which is any absence of identity. If everything that was other than something was a difference in the way we are speaking of here, then “inanimate” could be a difference added to animal, and non-being could be a difference added to being itself.
In general, to say that a genus is nothing other than a multitude of species or a species is nothing other than its individual members is to destroy the order of our knowing, which proceeds from the more universal to the less universal by various kinds of addition. This explains why St. Thomas did not need to argue for why the more universal could never be reduced to the more particular, for he saw that understanding the former was the very principle by which we understand the latter, and so to reduce the more universal to the less universal destroys the very means by which we understand the less universal.