Apparently, you are too young to remember the analogy of standing waste deep in gasoline with a fist full of matches. If using your match on your opponent means that you burn to death, then you are not going to light it. The term was "mutually assured destruction"
What you don't want is a lunatic showing up with his own matches. It's one thing for Russia, China, the US, etc. to have these matches, and another to give it to Iran and North Korea.
And disarming doesn't work, because as soon as one side thinks they could actually "win" a nuclear war, there is a chance that they would actually start one. But lunatics don't care if they can win, they are just insane, so we can't let them have them.
People complain about all the nukes, but how many launches have there been since WWII?
I'm not complaining about the fact there is nukes. Simply about the fact that only a select few have them.
Let's imagine, Russia who recently started a bunch of conflicts, is suddenly bombed, hard, by ISIS and decides to launch one at some opponent because they pinpointed their location and have to show they're strong and resolute etcetc. It's unlikely since Putin is many things but (as you said) a lunatic, but for the sake of arguments, say it happens (like he's getting old and senile and he has terrible counsellors).
Do you think someone else would use the nuclear weapon against them? Probably not, they weren't hit, and in this case there would be no escalation, and why should they start a nuclear war? There would be terrible consequences for Russia, but no matter what you do, a weapon would have been used.
Outside of that simple fact is also the problem of trusting current governments with them. Just because right now it's fine with who we have at the top doesn't mean it'll always be (and a case could be made Trump is treading a fine line with North Korea these days). Be it another guy elected or someone going crazy.
Hitler was elected, say someone like him comes to power again, how far would he be willing to go against Israel for instance?
Or even, imagine in a near future that a whole lot of countries decide to go against China/Russia/Europe because Hell, we're definitely evil in their point of view. They would have literally no opponent able to throw nukes at them either.
For a more tech point of view, take the Super-AI-gone-rogue scenario, calculating the odds of a nuclear response from potential ennemies and decining that anyway, it wouldn't be that bad to just wait a few dozen generations for the ultimate good of the human race.
Obviously all these scenarios are unlikely at worse but still *could* happen, just like the government could be going very much Big Brother on us with our Smartphones and stuff, or any other theory about what could go wrong in the future.
I'm just saying many things did go wrong during our History, and many things will still go awry, and these kind of power in only a few hands isn't necessarily a good thing.
Now, what could giving it to everyone brinbgs to the table ? Yes, more people willing to pull the trigger, but also more people able to intervene in case something goes wrong or if those who have it decide to ally. Which is, mutually assured destruction