This was a fun little rabbit-hole to jump into! My understanding is not that dissimilar from what you wrote in your comment:
"Basically IF Church-Turning & Brain finite THEN recursion."
But you forgot to add "and if there is arbitrarily deep layering of semantic meaning". So, really we would have:
If arbitrarily deep semantic layering AND the Church-Turing Thesis AND a finite brain, THEN recursion.
Now let's see if I can unpack that a little for you.
The authors are arguing that the components of sentences are placed into semantic layers that can group words rather differently than strict word-ordering would suggest, and furthermore, that these layers are effectively unlimited. Consider this sentence:
Mary and John went to the new movie (which was excellent) with Sam.
Mary, John, and Sam are at one semantic layer, the new movie is at a second, and the fact that the movie was excellent is at yet a third layer.
Mary and John went to the new movie, (which, as you expected, was excellent) with Sam.
We've now added a fourth semantic layer. And, I hope it is obvious that we could keep going! The sentence would become convoluted and hard to follow, but we could keep adding semantic layers as many times as we wish.
There is a second idea here as well. It is interesting that Mary, John, and Sam are at the same semantic layer, even though they are far apart in the sentence. This is evidence that there really is some kind of semantic layering, and not merely word proximity at play. We read "Mary and John", delve down into a new layer, and eventually return back to this first layer in order to add "with Sam". I mention this because it is already suggestive of recursion!
I understand your objection about having a finite amount of brain, and it is a very good counter-argument to the idea that we can follow arbitrarily deep semantic layers, but it is not an objection to the idea that human language is structured to be able to create them, which is really what the authors are arguing.
If you accept that language is structured in such a way as to permit this arbitrarily deep layering, then you are doing something very similar to a Turing Machine, which can represent any computable function using a finite number of states.
The Church-Turing thesis involves the notion that all of these representations are, somehow, recursive. This makes sense if you consider that all three of the systems involved (Godel, Turing, and Church) can do this same trick of going arbitrarily deep using a finite number of rules at the start. There must be some mechanism that allows arbitrary depth, which means that we must be able to repeat rules or states, which means recursion.
We are very close to having finished here. If arbitrarily deep semantic layering AND the Church-Turing Thesis AND a finite brain, THEN recursion.
We only are left with the finite brain. But of course, if the brain were infinite, then we would have no need for recursion. We could represent every semantic layer in a new region of the brain, and keep doing this forever.