The core arguments for South Korean nuclearization are well-known by now. Elsewhere, I have argued that there are two core drivers, which American bureaucratic resistance will increasingly find hard to ignore:

First, since 2017, North Korea has had the ability to strike the US mainland with a nuclear missile. This means that if the US should intervene in a Korean contingency, North Korea could strike the US with a nuclear weapon. This in turn might discourage the US from supporting its South Korean ally directly, per treaty requirement. This commitment credibility problem is a well-known issue in alliances. During the Cold War, France and Britain were so skeptical that the US would fight a nuclear war on their behalf (against the Soviet Union), that they built their own nukes. South Korea (and Japan) are increasingly in a similar position regarding Chinese and, especially, North Korean nuclear weapons.

Second, Donald Trump may return to the US presidency in 2025. He was noticeably cool toward US allies, especially South Korea, during his presidency. Indeed, Trump threatened to remove US forces from South Korea altogether if re-elected. This would almost certainly push South Korea to nuclearize immediately.