I don't think you have it right- the analogy to woodworking and craftsmanship is a category error and probably misses the broad thrust of the essay.

The goal of a woodworker or craftsman is the production of a finished good. He's arguing that, although it's been convenient to position a mathematician as a "theorem-producer", that's never really been the aim of mathematics, and that the actual products of mathematics are some kind of "mental software"- see his references to neuroplasticity. Basically, he's saying that the goal of mathematics is to create abstract structures that allow humans to reason about increasingly complex concepts, and that the "mathematician as theorem producer" is more like a convenient fiction that mathematicians have allowed to persist for too long, and now threatens to endanger the whole practice of mathematics.