And this is exactly the sort of thing that they don't show ingame but is the entire reason for the KanMusu's existence. When you look at it like this, Naka seems a lot more idol-like.
If all those idol anime have taught me anything, it's that those girls have to put in a lot of work into making those careers take off. Considering that she's first and foremost a ship fighting on the front lines, all her idol work is something she can only do in her spare time.
Say what you want about Naka, but at least she gets points for dedication.