I don't see the author arguing that being open makes software inherently more secure, but that your software shouldn't rely on being 'secure' because it's closed source, rather design the software to be secure even if the source code were to be open-sourced, which may bring additional eyes, (and if not, it's no worse than if it was closed).

I don't know if anyone has a good argument against that, but I didn't see one yet. It's not an argument to say that open-source makes your software bulletproof or even safer just by being open-source, rather if you designed your software with proper security as a goal, you shouldn't mind releasing your code.