> In the case of human spoken language, the recursion exists in our characterization of the grammar
But people are constructing sentences, not grammars. When you construct a grammar you can add a recursive part to it, that is true, just like in a programming language, but constructing grammars is not what people mean with language skills.
A sentence can't be recursive since languages in themselves has no concept of applying a concept, for that you need an interpretation of the language references. For example, you can have a recursive function written in a programming language that doesn't have a recursive grammar, the concepts are different things.
There are two ways that recursion intersects with language that are relevant here:
1. Our spoken and especially written grammar is recursive. We do handle this unconsciously. This is not related to our ability to reason about recursion at a high level, and recursive grammars are not necessary to do so. This is not a skill in the normal sense and we have only (very) limited ability to improve our capacity to interpret deeply nested grammars. However, this is still a useful illustration of what recursion IS, which is why I brought it up.
2. Language also introduces the ability to semantically reason about recursiveness. This is still a linguistic thing—you need a symbol and relations among symbols in order for recursion to be meaningful—but this is a skill and is likely very related to linguistic skill. This is the part that really helps you to program: ultimately, you're just reasoning about symbols and looking for incoherency.
But people are constructing sentences, not grammars. When you construct a grammar you can add a recursive part to it, that is true, just like in a programming language, but constructing grammars is not what people mean with language skills.
A sentence can't be recursive since languages in themselves has no concept of applying a concept, for that you need an interpretation of the language references. For example, you can have a recursive function written in a programming language that doesn't have a recursive grammar, the concepts are different things.