author here. the idea is to see how the Y and Z combinators come out relatively naturally given an extremely constrained context with no loops, no recursion, no bindings. the article is intentionally structured as a sequence of failed attempts that progressively reveal which phenomenon is still responsible for the recursive behaviour.

in the article, i define recursion narrowly as "... call the function `fact` from inside the definition of the function `fact`".

you're right that `factgen` is also not recursive but it is also rejected, but for a different reason: self-reference. declarations are banned precisely because they give a function a "name" that can be referenced later.

please note that the solution which uses Z combinator (given at the end) grows from `(x => x(x))(x => x(x))`, which is NOT self-referential. it achieves self-replication by literally rewriting the content of the function twice, which is different from self-referential recursion of `factgen`.

hope that clears some of the confusion.