I too have in my early days of learning, struggled with such an objection. I think this objection stems from a lack of understanding of the actul argument. When Feser lays out the argument, he starts off with the existence of change; however, such a rhetorical move is only made in order to establish or attempt to establish the metaphysical principles act and potency. But it is not from this which he derives something that is pure act. It is the sustanence of potencys in act rather than the temporal actualization of a potency itself. For example, it is the refrigerators ability to continually actualize waters potential to be ice, rather than its ability to actualize what was once water into ice.
Now having understood this, that it is the continued actualization of potentialities rather than temporal ones, the obejction fails. If, for example, we arrive at the existence of the ontologically absolute (Which is in fact what the argument arrives at), then your objection could be stated: but why must this thing be immaterial, divinely simple, and the like? It could just as well be composed of parts but have no potentialities to actualize. But this is mistaken. Continually actualized potentialities exist in the composed, which is what the argument is seeking to eleminate; the composed wholes potentility to be whole is itself continually actualized by each part, and is therefore not really pure act; there must be something even lower, so to speak, which holds this in existence. If one follows the logic, you will arrive at something which is pure act. Not something which just has no capacity to not exist, but something which has no capacity to not exist and whos capacity is kept in existence in terms of "itself", rather than something else. Your argument would leave you with something logically necessary and of derivative existence, but the argument actually leads you to something deeper: something logically necessary and something which is of underived existence; something truly pure act.