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Nuclear weapons are not a panacea. They do not replace a conventional deterrent. If all you have are nuclear weapons and nothing else then your enemy has escalation dominance and can keep pushing your red lines bit by bit. Think about it, are you going to respond with an all out nuclear attack to every provocation? Or are you going to let the enemy employ salami slicing tactics until they are in a position to take out your nuclear deterrent and leave you with nothing?

Moreover, a country with nuclear weapons but a weak or nonexistent conventional military is effectively inviting a first strike on its nuclear capabilities because if those get neutralized the enemy then knows nothing can stop them. We know that the US' leaders are increasingly irrational and delusional. What if they decide you don't have the guts to actually pull the trigger on nuclear retaliation for a limited incursion or bombing campaign? What if they think they can take your nukes out before you launch, or they convince themselves that your weapons don't work or that they can defend against them?

Another issue is that simply having nuclear weapons is not enough, you also need to be able to deliver them on target, and in sufficient quantities to make the enemy pay more than what they consider an acceptable cost for destroying you. The DPRK don't have nuclear submarines and they won't ever gain the aerial superiority required to deliver nuclear payload by bombers so they are left with only one leg of the nuclear triad which is ground launched missiles. And those can be intercepted, especially if the distance is long as it is to the US mainland and the enemy has a large military presence in between, which the US does with its navy and its many bases in the Pacific. And the DPRK don't yet, as far as we know, have the hypersonic technology that Russia has which would make interception much harder. They also likely don't have thousands of nuclear missiles so they can't just rely on sheer numbers and betting that enough will get through to cause significant enough damage.

It would be one thing if they were only facing the puppet regime in occupied Korea, they can more than likely level all their cities as the distances there are too short to intercept, but their real enemy is an ocean away, with a large territory and forces spread out all over the globe. The only real way to guarantee your safety is for the US to be aware that you have the capabilities to fend off at least partially any first strike attempt (i.e. you have a good integrated air defense), AND that even if nukes are never launched you can make any potential conventional war very unpleasant and costly for them.

Admittedly this is a big cost to pay for a small country like the DPRK, but unless you have a bigger country to protect you it's either that or inviting destruction.