Sounds as if you know SK’s status pretty well.
Why are we there ?
Primarily, in order to show both of the Koreas that in the event of a war between the two, the US will automatically join with the South. That’s the reason what ground troops we do have are positioned so close to the DMZ. Militarily, positioning them there is a bad idea. But it serves a political purpose - by putting them directly in the line of fire if war breaks out, it sends a message that the US would be drawn into said war - we wouldn’t really have the option of deciding to ignore any defense treaties we have with the South.
South Korea is more then capable of defeating the north militarily on its own. But that would come at the cost of grievous casualties to the civilian population due to the truly immense numbers of artillery pieces the North Koreans have along the DMZ. Even without nukes, they could potentially kill hundreds of thousands of people. US air and naval power would be the primary contribution in the event of a war - knocking out North Korean artillery and rocket batteries.