I think that the denial of an infinite regress in cosmological arguments is self-evident (since an appeal to infinite causes no more explains the effect than the stability of the earth can be explained as the effect of turtles all the way down) but it’s not necessary to appeal to this denial to make a cosmological argument. Avicenna, for example, replaces this step in the argument with what might be called an appeal to the insufficiency of the totality, regardless of whether the totality is infinite or finite.
Consider the First Way, which starts by observing the existence of motion among moved movers. Next, rather than denying that a series of movers can be infinite, instead consider the totality of motions by moved movers. This totality is either caused to move by the whole, or by some part. But if some part moves all the others, then that part is an unmoved mover (which is impossible, since it is one of the totality of moved movers) and if the whole is the source of motion, this either means every part is an unmoved mover (same problem as before) or we are talking about the whole in opposition to all of its parts; but there is no such thing as a whole in opposition to all its parts.
Even if one took issue with the reasons given, the basic sense is to consider the totality of all things established at the first stage of the cosmological argument, and then to say that the totality, whether considered as all its parts or some of them, cannot account for what is observed at this first stage.