If they went to war with the North, and won, they would have to deal with a whole new population of unknown numbers that are starving and largely brainwashed and/or ignorant of things outside of their nation.

Western Europe, the UK, and the US happily take in similar people but I think South Koreans actually understand the chaos this would create. Just building basic infrastructure for an entire country like North Korea would be extremely difficult without large amounts of outside funding. Only China and Russia would be willing to do so but they both use NK for what is essentially slave labor. So they have no incentive to actually do it.