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?
It's the same outcome either way. You don't have nukes and another country decides to nuke you? Your country doesn't exist anymore! You do have nukes and another country decides to nukes you? Your country doesn't exist anymore! What changes?
People say deterrence, but what is the deterrence? You built something that you'll never use? What's the point?? Oh you will use it? Great! You've decided there's some event that is so bad you'd end the world if it happened. I'm not sure what event that is. Maybe you have one in mind? China attacks India? The world should surely be destroyed then! No? Too bad! You don't get a say! China and India decide if humanity gets to continue! They definitely wouldn't do that though.
They built their nukes to never use them. Which is the same as not having nukes, but having nukes is required so that nobody uses them, which is the same as never building them, but they need to be built so they won't be used!