Which of my claims is incorrect?
(To clarify: I am not disputing the physical principle. My claim is that life is not more physically “complex” than a singularity or the Sahara Desert.)

Life is far more complex. Incomparably more. But I can’t present a calculation here. Notice that you didn’t present one either. Intuitively, think about the transition from the Sahara Desert to producing a living creature. That is very improbable. But for a desert to arise from a living creature—that is possible.

I agree that life intuitively feels more complex, but the Rabbi’s whole innovation was to give that claim an objective meaning by means of entropy. In my opinion, that doesn’t hold water. By the way, when you talk about what can produce something else, you’re changing your definition of complexity; that is not necessarily connected to entropy.
Why doesn’t the physico-theological argument go like this: a world that begins in a state where all matter is concentrated at one point is very special and complex (very low entropy). Everything complex has a composer, etc.?

First of all, above you wrote that you think the opposite—that the desert is more complex. That is incorrect.
As for the matter itself, you are mixing together two planes. I did not bring up entropy in order to present quantitative measures of complexity. I brought it up in order to reject the claim that complexity is a subjective matter. The complexity of life is objectively higher, and that is a scientific fact. To arrive at an exact number, one would have to do a calculation, and this is not the place for that (I also do not know the details of the calculations, but they do exist. There is even a calculation of how low the entropy in the Big Bang had to be in order for life to arise).
The argument you brought at the end is really the same argument itself: complex things came into being, therefore there is a creator. It is just that life is the most complex thing, so it is the clearest thing to point to. But there is no principled reason to focus דווקא on life.

What I meant was that if you define complexity by entropy, then it follows that the desert is more complex (it has lower entropy), and therefore I do not think entropy is a good definition of complexity in the way we usually relate to it. As stated, I am not disputing the physical principle; I am only saying that it does not match our intuitive understanding of complexity.
I didn’t really understand the bottom line: do you define complexity by the size of the entropy (that is, lower entropy = higher complexity)? And if not, then what is your definition?
(I saw an interesting definition that in my opinion is better: complexity is a balance between simplicity—or uniformity—and chaos.)

Entropy is an excellent measure of complexity. I would not go so far as to establish it as the definition of the concept. I have also written more than once that the physico-theological argument does not rely on the second law of thermodynamics (because otherwise I would have to assume that it is violated again and again throughout history). Moreover, the law determines the framework within which everything operates, but the argument deals with the framework itself (what I called the argument from the laws, not the argument within the laws).