Classical Theism

I have been scouring the internet for quite some time now searching for an answer to this objection. Looking at the archives of the old CT Forum the objection was brought up there as well, especially by RomanJoe, but never was a satisfactory answer given. The objection is this: from the premises of Feser's Aristotelian Proof (potentialities are actualized, an essentially ordered causal series must have a first member, a potency can be reduced to act only by something that is in act) we cannot deduce the existence of something that is purely actual, only the existence of an unactualized actualizer which exists in an underived way (nothing actualizes its existence).

Whenever this objection is brought up, the answer is usually that if this unactualized actualizer were to have any potencies, it would be a composite, and a composite cannot exist in an underived way, so the unactualized actualizer has to be purely actual. Sometimes the answer given is that if the unactualized actualizer had a distinction between its essence and its existence, it would not exist in an underived way, so its essence and its existence have to be identical, and such a thing has to be purely actual. I would certainly agree with both of these assertions. However, the point is that from the given premises of Feser's Aristotelian Proof, we cannot arrive at that which is Pure Act. Thus the argument fails as a standalone argument for demonstrating the existence of God; to arrive at God, we have to appeal to further premises, premises that are not part of the actual argument.

Am I mistaken? Can we actually arrive at the existence of Pure Act from the three premises of the argument?