While the structures-that-be were woefully unprepared for the ferry disaster and in fact were responsible for the existence of the factors that led to it in the first place, I don't think the president is directly responsible, and her part, for the micro-level incident that was the result of many macro-level factors. The main difference here between this incident and something like the hurricane Katrina disaster in the US lies in the fact that the federal government in the latter case had ample warning and simply chose to not prepare for the very eventuality that experts were busy warning them about. And then the rescue effort itself, I don't think anyone believes that the United States did not have the power or capability of rescuing certain victims, but was simply uninterested in doing so. What I would blame the United States for is a willful disregard for black lives, while I think the South Korean president was simply in the middle of institutional structures that were woefully unprepared for the disaster that came into their laps and is, at most, guilty of being at the head of a structure that was incapable of anything more then incompetent bungling. Again, it's obvious that Park is catching a lot of heat because people are politically dissatisfied with her. And the ferry incident only adds fuel to that dissatisfaction. But I don't think she is, in any moral or ethical sense of the word, directly "responsible."
In this sense, I see her crying as a bit more sincere than the crocodile tears shed by then former Korean president, Roh Tae Woo, in 1996. I think Park is doing the Korean thing and stepping up to the plate and offering herself as the symbolically responsible and culpable leader, since people want someone to blame. But that's all she can do since as a president and a Park, she won't resign. Her family doesn't have a history of giving up power easily, as we all know. It's a very old-school Confucian thing. The crying is expected. Noh Tae Woo was a lying snake. He could have cared less about the morality of his deeds. Yet crying as an act of penitence was expected in his role and circumstances. Same for Park now. And not the same for Hillary Clinton back in the 2008 elections, which is when and why I lost all respect for her. Hilary isn't the type of person to break down and cry, so I saw this as a calculated political ploy to gain votes when her numbers were looking dangerously bad. Both Clintons were known for crocodile tears, which as an American, I find merely insincere and completely out of character for them.
I think in the traditional Confucian context, it's not so much the sincerity of the act, but merely that the act is expected given the position and circumstances.