As others have already pointed out the many other flaws in this argument - the fact that the 'designers' would fall into the same set as all the things they deigned and therefore create a Russelian paradox, We have no example of a thing that has not been designed in the universe so the argument is unfalsifiable - I will simply try to explain the set theory argument in clearer terms.
To start, there must be two sets A{all things where the whole exhibits a higher order function than the parts} and B{all things where the whole does not exhibit a higher order function than the parts}.
Many human creations are in set A, a pile of rocks is in set B, all animals are in set A, many human creations are in set B (humans could pile rocks without any design or purpose). So at the moment we have the observation that some of the things in set A are designed (by humans), some of the things in set B might also be designed (it is possible that a human deliberately piled rocks with the intention that they should serve no purpose other than to look like a natural pile of rocks).
No matter how many human devices there are on the planet, they will be outnumbered by the weight of animals - bacteria, beetles etc. So as things stand, all we can infer so far is that a small proportion of the things in set A have been designed, as have some of the things in set B.
Your argument then goes on to say that because a small proportion of the things in set A have been designed, it is logically compelling to presume that
all things in set A have been designed. Of course, a small proportion of things in set B have been designed too, so the argument must be applied there also. Thus the argument dissolves to - because some things have been designed, all things must have been designed (seeing as set A and set B together comprise all things). This is obviously nonsense.
This is the problem with inferring (without cause) that all members of a set share all the same properties as all other members of a set. This is simply not how logic works. The
only things members of a set can be logically demonstrated to share is the
one characteristic that makes them members of that set. We could, using Bayesian inference, say with increasing certainty that all members of a set share a non-necessary property as the number of members that share that property exceed half the set. If more than half of all people called John turn out to be clever, we can begin to infer with increasing certainty that all people called John are clever.
But human creations do not come anywhere close to half of set A, the number of bacteria alone outnumber human creations by whole orders of magnitude, not to mention the fact that once we know a mobile phone is the creation of a designer, a further mobile phone does not become further evidence of this theory.
So all we have in terms of data is still that a very small proportion of all things in set A are designed. By what logic do we then presume that it is even likely, let alone inevitable, that
all things in set A must share this same property? We know it is not a necessary property (as has famously been referenced, if you assemble all the parts of a watch randomly, with no design, in a different order over and over you will eventually assemble a working watch). We also know that a small proportion of things in set B are designed, which together comprise the set {all things}. So we are left with no reason to presume that all the things in the set share the same property as some small proportion of the things in the set.