It’s not like he’d fail the bridge officer’s test.

He was clearly a qualified bridge officer. As a member of the regular chain of command (and being the one regularly in charge of the night shift due to not needing to sleep), he had to be. (In fact, he was put in command of another ship once which would have been impossible if he didn’t have the qualifications.) In his case it’s probably a lack of ambition that led to him being stuck in that rank; he had no real desire to be promoted to another ship when his friends were on the Enterprise and he had every opportunity to learn about meatbags there.