Problem is not that China would - it wouldn't.Maybe but what if then China replaced SK and US in your sentence with Cuba and China?
Problem is nuclear tech was 'rocket science' in 1940s. Making a bomb in 2020s isn't that hard anymore, nor - as rocket startups show, - is making an ICBM.
Blink and you'll enter a world where any technologically-weaker opponent relies on MAD deterrence.
And, if it isn't enough, then just wait before someone will press the button.
If Chekhov's gun remained silent between few sane, cold-minded actors for surprisingly long time - then let's issue it to everyone, what can go wrong.