The Mathematical Reason You should have 9 Kids

He's referring to biological value, as it's defined in: "What is value?". Biological value is not the same as the type of value that you're thinking of. It's distinct from the other types of value. Biological value claims are truth claims, unlike other types of value claims. A claim about biological value, such as "cutting down the tree is bad for the tree" or "overdosing on fentanyl is bad for you" is a truth judgment, not a value judgment. I could want to cut down the tree, and still understand that it is bad for the tree to be cut down.

Not only weren't organisms created to reproduce.

Organisms are designed to reproduce. If organisms weren't designed to reproduce, then they won't reproduce, they will have no descendants, and organisms will stop existing altogether.

Organisms just do what they do. They exist because certain structures tend to reproduce themselves, and those structures can occur naturally.

Yes, exactly.

It's dangerous to do even that.

No, it isn't. It's impossible to describe biology without using normative language, since biology is intrinsically teleological. I don't understand why this is a stumbling block. We always use normative language when talking about biology. Normative language is often used descriptively, as in "this soil is bad for pine trees". Or "this is a good hammer" (even if you have no use for a hammer). Disease is "dis"-"ease". Disorder is "dis"-"order". There is no way to talk about biology without using normative terms.

Evolution selects forms based on their effects. Thus, the effects explain the form. But it's not just the effects. Certain effects, which might not even be probable, explain the form. Those effects are the telos. The telos explains the form.

You have to be careful to remember that the word "purpose" there is a metaphor.

I really don't understand what your issue is. He said "biological purpose", not purpose in general. The author understands that biological purpose is not the same thing as subjective purpose. These two types of purpose are clearly different concepts, and I see no conceptual mess. There is no ambiguity here.

You misunderstand that paragraph. I'm friends with the author, and he doesn't believe in objective morality, nor does he believe that it's "morally correct" to reproduce. Replicating a genome implies reproduction (unless it's the genome is being artificially created through cloning), but reproduction doesn't necessarily imply replicating a genome. For example, if you reproduce with someone who has very different genetics (i.e. someone from a different race), then half of the offspring's genome would be quite different from your genome, compared to if you reproduced with someone from the same race as you.

He does believe that organism's biological purpose is to reproduce, but that doesn't mean that he believes that organisms should reproduce. It's up to the organism whether it reproduces or not. As he said at the beginning of the paragraph: "I do not believe there is a correct number of children to have". From that statement, it's implied that he doesn't think it's "incorrect" to have no children at all, so I don't understand why you concluded that he thinks that it's "morally correct" to reproduce.

"An organism's biological purpose is simply to reproduce" is a truth claim, not a value claim. The only value claim that he stated in that paragraph was "I do not believe there is a correct number of children to have".