Yes, he could easily have told his chain of command what they were doing, and why they were doing it. If it was legitimate, the higher-ups would have said, "Ok, do what needs to be done and keep us posted." The ship fixed it and got underway again in a few hours, so then he could have said, "We're good to go now, and back on schedule. Full details to follow." End of story. But it didn't happen that way.
Unless he was trying to hide something, why else would they do this sort of thing? For example, was this propeller issue related to something they'd had previous issues with, and that was supposed to have been repaired? Would the chain of command think badly of this Captain for this issue occurring again? Did it happen due to something negligent he was doing with the ship? Did it stop working correctly due to something this Captain was responsible for? No way of knowing any of it. But for him to try and cover his tracks by hiding the ship's operational status like that just seems really weird. Navy ships and Navy Captains just don't DO that sort of thing. Satellites and submarines track and report ship movements, and somebody somewhere would have surely caught on to things being out of sync. Then he'd have even more explaining to do. So the reasons why he did this aren't easily explained. As I said, there is more to this story than is being reported. We also may never hear the whole story.
Dave