What doctrine is this that elicits such strong rhetoric? Perhaps you think the answer has something to do with Jesus Christ, or a major biblical teaching. Surprisingly, the answer has nothing to do with either.
Anyone who does not know that proponents of divine simplicity take it to be directly connected with both the doctrine of creation and the doctrine of the unity of God, both of which are very obviously major biblical teachings, and have consistently done so on both the Jewish and the Christian sides for well over a thousand years now, does not know enough about the doctrine to be talking about it. (The "strong rhetoric" to which Mullins refers is the claim that denial of divine simplicity leads to idolatry or atheism if consistent. He makes it sound as if this were just some arbitrary thing people said. Did it never occur to him to consider why anyone would say that?) And what is very, very noticeable that nowhere in his essay does Mullins at any point do what should be the first step for any serious criticism of a major position: he never looks at why the doctrine is held in the first place. The closest he comes is trying to tie it, very vaguely and noncommittally, to perfect being theology -- which is not at all a standard reason for it. The doctrine of simplicity did not become a major position because people said that God is a perfect being and then said, "And I guess one model you could have of a perfect being is that it's not composed out of anything more fundamental." Indeed, talk of God as "perfect being" is a very, very late development that originally presupposed the doctrine of simplicity; there is no word in Greek or Latin that is the direct correlate of our word "perfect", although there are circumlocutions for things that we might classify with that word. "Perfectus" in Latin means 'complete', or more strictly, 'completed'; this was also one of the original meanings of the word "perfect" in English (technically still is, although mostly confined to certain kinds of longstanding expressions). When Aquinas in the thirteenth century asked whether it was appropriate to call God perfect, he had to draw a very careful distinction about the different things you could mean if you said God was complete, and a key step to ruling it admissible was that it could not be taken to mean (as it would usually be taken to mean in applying the term to creatures) that God could also be incomplete in the sense of partial, because that violated the doctrine of simplicity. We call God 'perfect' because there was a sense of the word consistent with the doctrine of divine simplicity, not vice versa.
So Mullins is starting entirely the wrong way around. He should instead start with the doctrine of creation. God is the uncreated Creator of other things. It is this that serves as the foundation for the theological doctrine of simplicity, and the basic line of thought is that given by the great Saadia Gaon: when we look at creatures and ask what in them indicates that they were created, shows their status as creatures, we do find such indicators. These are things like mutability, dependence, and composition. So God, who is uncreated, must lack these telltale markers of things that are made, and thus must be noncomposite -- simple. In addition, the term 'simple' is historically a relative term, in the sense of admitting of more and less; you can find any number of people arguing that the soul is more simple than the body, for instance, or that the saints by divine grace are made more simple, all in this sense of noncompositeness; the idea is that the soul is more unified than the body, less divisible into parts and therefore more properly called 'one'. And since God has none of the divisibility into parts, none of the composition that indicates createdness, He is least composite and therefore most simple.
That is it. That is the essential idea of divine simplicity. Mullins makes a very big show of how complicated and difficult the doctrine is to understand; this is obvious nonsense, as you can find laymen in practically any large Presbyterian or Catholic church who can fully and completely understand the point that God, being unmade and unmakeable, is not in any way made out of anything. Complications only arise when you are no longer asking what the doctrine of simplicity itself means and start asking how this or that already complicated topic relates to it. It's not an accident that Mullins runs to talk of properties: there is no generally accepted theory of properties, literally none at all, so any discussion of properties is necessarily complicated in order to pin down what you are talking about; therefore any discussion of simplicity and properties is necessarily complicated. But the doctrine of divine simplicity is not in any way downstream from any account of divine 'properties', in whatever of the many, many senses of that term you are using it. It's likewise not surprising that he builds another criticism out of the application of simplicity to freedom and necessity; these are very complicated topics. There's nothing wrong with seeing what a simple position implies about a complicated topic; but it is very, very absurd to complain that applying a simple position to a complicated topic gets complicated.
Another example of complicating the basic position by applying it to something complicated is his reliance on the notion of identity. Identity is a notoriously difficult subject in analytic philosophy; of all equivalence relations, identity is the one we least understand. It is also a concept that is fairly new. Mullins says:
On the classical understanding of God, theologians will say that all of God’s essential properties are identical to each other, and identical to the divine nature, which is identical to God’s existence. The identity claim here is very strong, and can be easily missed. This is because we use the word “identity” in rather loose ways in contemporary English.
This is not "the classical understanding of God"; this is a translation, a reconstruction, of the classical understanding within a specific vocabulary, that of analytic philosophy. And the problem is that it seems to be based on an assumption that the Latin word "identitas" means "identity" in the analytic sense. This is a very false assumption, because "identitas" is a looser and weaker word, not a stronger and stricter word, than the word "identity" in colloquial sense. It just means 'sameness', in most of the senses we would give the word 'same'. The primary application of the word is in saying that things are the same kind of thing, but it can also cover other kinds of sameness. When people did theology in Greek and Latin they had no word at all for the "very strong" sense of identity to which Mullins is pointed. They could talk about it, but it required some complicated circumlocutions. And it was not the sense in which 'sameness' was used when talking about divine simplicity, in saying, for instance, that in God wisdom and power are the same. (Aquinas, for instance, pretty clearly denies that the "very strong" sense of sameness is the right one in this context.) One of the most influential texts in the Latin West on discussions of the doctrine is Augustine's De Trinitate; in his brief comments on the point, Augustine's analogy for divine simplicity is the unity of the virtues. In a fully virtuous person, even though "prudence" and "justice" and "fortitude" and the like are not synonymous words, justice will be prudent, courageous, etc.; this way in which all the virtue-terms applied to a fully virtuous person in some sense include all the others is the closest we come to something like divine simplicity, the main difference is that we can't be prudent, etc., except by acquiring these bit by bit, whereas for God, to be God is already to be wise, good, etc. Try to translate the unity of virtues analogy into standard analytic identity-talk and you get gibberish. That is a warning sign that the new terms are bringing baggage with them that the original terms might not carry.
In any case, Mullins's divine freedom argument against simplicity depends entirely on the "very strong" sense; Premise 8, for instance, requires strict transitivity, despite the fact that, historically, theologians and philosophers have denied that strict transitivity applies to the kind of sameness talked about in divine simplicity. Mullins does suggest that certain common arguments require it, but I think a closer examination of those arguments than Mullins gives would show that (1) he is confusing general implications of simplicity itself with transitivity of specific properties; (2) he is dropping, as he does all the way through, the notion of compositeness as a sign of createdness; and (3) the arguments generally have perfectly acceptable analogues in Augustine's virtue analogy despite the latter clearly requiring the rejection of strict transitivity. To put it in fashionable theological terminology, all one requires for a doctrine of divine simplicity is 'perichoresis'; but perichoresis does not imply that one thing can always be directly substituted for another in every context.
While not my main point, it's perhaps worth noting that even assuming the application of identity, the argument is not as straightforward as he suggests, since it is known that even the strong form of identity, even in ordinary cases, only allows intersubstitution within the same modal context; if we are talking about things that are described in different modal contexts, things get immensely more complicated, and you can't assume that something like Premise 8 would apply. And as it would be obviously false to say that terms like divine necessity and Creator have no modalities, and very implausible to say that they share exactly the same modalities, you would need to establish that claims about them could be formulated in the same modal context before you could use a premise like Premise 8. Otherwise it would be exactly like claiming that, because the real temperature measured by a Celsius scale and the real temperature measured by a Fahrenheit scale are exactly the same real temperature, therefore Celsius scales are Fahrenheit scales. 'Real temperature measured by a Fahrenheit scale' and 'real temperature measured by a Celsius scale' are modally different descriptions. Mullins completely muddles this in his discussion of what he calls the "Modal Mystery strategy", assuming rather than establishing a unitary modal context that makes it so that the modal term 'necessity' is not shifting meanings in the different uses.
Mullins also obscures the matter by ambiguous use of terms. The puzzle about freedom that he tries to develop can't arise if we are using 'freedom' in an intransitive way: 'God is free'. There is no problem whatsoever with claiming that this is necessarily true or that God is necessarily free. The puzzle only arises if we are talking about freedom transitively so that it is taking a non-necessary object. But while it is obviously the case that God-freely-doing-this-non-necessary-thing is not necessary, this is obviously because 'God-freely-doing-this-non-necessary-thing' is a mixed description that depends on something that's not God. 'God being free' and 'God freely doing X' do not share exactly the same modal context; the one is entirely about God Himself, but the latter is about God only relative to X. "1+1=2" is necessary, but "this apple I am picking up and this other apple I am picking up are two apples" is, strictly speaking, not, because it is neither necessary that there be apples nor that I pick them up; it is a mixed description that partly depends on something that's neither a 1 nor a 2 nor a mathematical operation. It would be absurd to claim that talking about necessary numbers in terms using contingent apples makes numbers contingent or apples necessary. And equally obviously, a description that includes a creature is not intersubstitutable with a description that does not.
These are different versions of points that are also made by both Lenow and Feser in their contributions.