While it is true that some rules of English state that “he” is gender-neutral, in practice this is not the case.
“If the customer needs to use the men’s bathroom, he should request the key from Alan. If the customer needs to use the woman’s bathroom, he should request a key from Sandra.”
I have never seen anything like that written as instructions. When we say “he was a doctor” or “he was a race car driver”, our listeners and readers do not hear about a non-gendered person. They hear us talking (or writing) about a male.
In my opinion, usage of English shouldn’t change when it conflicts with the ‘rules’. The rules were written down by people who were either trying to write down how English is used, or by people who wanted to force English into how they wanted it to be. English does not follow rules. Rules describe English. We forget this at our peril. Where rules and the actual use of the language collide, the rules are clearly wrong.