A general, logical way of analyzing that argument would be,
i) the less random something is, the more non-arbitrary it is, and the more information it contains. (This is a common idea from information theory.)
ii) Something non-arbitrary is something borne of logical structure. The more constrained a set of relationships, such as propositions “following” from each other, the more that something attains what is commonly caused “causational” aspects (which, as I currently view it, is just the subset of “physical” propositions of logical consequence, from the enclosing set of all chains of logically consequent truths.)
iii) Therefore, if something appears non-arbitrary, then it must have highly particular logical, causal principles by which it came to be.
I have more to say on this. But for now, I believe the “fallacy” is about an interesting intersection of observation and observer.
Perhaps, to give one example: if the “observer” (say, a sentient being) has the scope of their observation ability delimited in a particular way, it can create the illusion of non-randomness, but that may be a small pocket of order in a broader context of randomness.
I believe in psychology this may be called “cohesion”: things only manifest on the horizons of our conscious, cognitive apperception, which are predetermined to do so, due to having some characteristic which “gets them in the door”. Thus, the “dynamic” between observer and observation I mentioned above, is that they are causally/logically linked: only that which the human mind, being situated in highly particular circumstances, predetermines to be made observable, becomes observed.
I am willing to go out on a limb here and say that my idea is not optimally developed or clear, but intuitively, I feel there is something strong here I am getting at. On a far deeper level, it is about things which we do not realize are far more profoundly causally and inherently logically linked, than we realize. We sometimes find the coherence between things in our universe to be deep, because, with our limited point of view, we do not realize that two separate things are so intrinsically logically linked to one another, so as to be fully identifiable with one another; and I think this is a very Wittgensteinian idea.
We can’t fathom how two seemingly different things could be connected, because we do not grasp that the two separate things are actually deeply interconnected to the point of being mutually necessary or dependent, or even essentially 100% mutually causally linked.
I will try to develop what I am saying in a more formalized way.