You make a significant error in pairing Chinese strategic interests with North Korea's. Even the available open-source information indicates that their relationship is far more complex, complicated and increasingly at odds.  You also make an incorrect assumption that a very limited North Korean nuclear capability can't be countered. 

No liberal democracy would choose to start to start the shooting given the immediate stakes in civilian lives, on their own side. Containment, in whatever form it takes, is the proper solution. Under any scenario, DPRK is not an existential threat to the US or even the ROK for that matter. If they pull the trigger they get one shot, it's likely to fail and under any circumstances they will not survive the consequences.