No. Nuclear weapons should not exist.

Kurzgesagt recently made a video on the nuclear arms race. The end of the race was when the guy who invented the hydrogen bomb invented a bomb that could destroy the entire planet. The bomb wouldn't even need to be dropped onto your enemy. It could be built inside your own country and detonated any time at all to end humanity. He thought of it as the biggest deterrent to war. Nobody else did. Politicians and military leaders threw out the idea entirely. Why would anyone detonate a nuclear bomb inside their own country??

The size of that bomb pales in comparison to the size of all nuclear weapons in existence today. We built that bomb. It's just not one giant bomb, but split into 12,000 parts and spread over the world. Is it any different? If you cannot justify building a nuclear weapon that would destroy your own country to destroy another, how can you justify building any nuclear weapons at all?