Why doesn't South Korea give up its claim on North Korea?
Let me ask a rhetorical question. Why doesn't the north give up its claims on the south?
By now it's completely obvious that the Korean peninsula won't be reuniting any time soon.
No it isn't. Obviously Germany split around the same time as the Koreas but still reunified. If there is any angst about that, I haven't heard it.
Beyond the question of whether or not the community wants to reunite, there is also the constant possibility of military action. The United States is not going to rule North Korea under martial law. If the United States goes to war with North Korea, the only practical end (assuming a US victory) is for South Korea to take over in the North. And if South Korea does that, their own constitution would require that they allow the north's citizens to vote. They'd be reunited.
Personally, I would expect them to reunite by the end of the century. Obviously North Korea's model is not sustainable. If it was, they wouldn't have to sell chemical weapons to Syria just to keep the lights on. The Koreas will either find a way to work together, which will inevitably to reunification, or North Korea will collapse, which will inevitably lead to reunification.
The only path to keeping the two separate that I can see would be for China to manage North Korea as a client nation. And China doesn't seem very interested in doing that. I suppose Russia could take the place of China if they wanted, but I'm not sure that anyone wants that either.