What about US nuke cover for South Korea?

In all the euphoria about South and North Korea agreeing to denuclearise the peninsula, what is Kim Jong-un's approach going to be towards America's nuclear weapons? There haven't been any US tactical nukes in South Korea since they were all removed in 1991. But the US provides a nuclear umbrella for South Korea, principally with its airborne and submarine-launched nuclear ballistic missiles. Under the longstanding security pact btween Washington and Seoul, were South Korea to be targeted with nuclear weapons from Pyongyang, the president of the United States would press the button and launch similar type nukes aganst North Korea. That's what nuclear umbrella means. Japan and Europe enjoy the same nuclear insurance policy. It's not quite the same as having nuclear weapons actualy sitting on South Korean soil but the deterrent value of the umbrella is as effective. Kim knows this and he will want some quid pro quo if he really is contemplating getting rid of his nukes. Like, "remove the nuclear unbrella, Mr President." This is just not going to happen. Well, not in anywhere near the forseeable future. As I have written before, Kim will want to sit down with Trump as equals in the nuclear weapons business. Why would he scrap his huge nuclear weapons infrastructure in return for some promise that the Americans would no longer maintain a nuclear umbrella over South Korea. Bizarrely, it might have been an easier negotiation if the US DID still have tactical nuclear weapons in South Korea. Then Trump could say, "You remove yours and I'll remove mine". Just like John F Kennedy did when Nikita Krushchev finally backed down and removed the Soviet nukes from Cuba in 1961. Kennedy agreed to withdraw US nukes from Turkey. Trump has no such easy bargaining chip. He will have to offer Kim something in return but it's going to have to be pretty substantial to persuade Kim to get rid of ALL of his nukes, unless the North Korean leader has really had a Damascene moment and no longer thinks nuclear weapons have any purpose at all. As far as I know, Kim has never been on the road to Damascus, so....