Is that how a [however pseudo]military would operate? I would think they would use the "chain of command" that they even reference in the show. Perhaps I misunderstand, but I understand the phrase "chain of command" to mean one person gets the talking-to from the boss and then has his underlings get it done.
But remember this is a ship full of enlightened Roddenberry super-humans. They don't need a "talking-to".
Presumably the "chief engineer" is the senior officer on duty at the time. If you want something done, to tell them to do it and they co-ordinate with their staff. When Argyle's shift ends, he hands over to MacDougal. One would hope they were competent enough officers to be able to continue a project after a mere shift change.
Even Geordi wasn't on duty 24 hours a day.