Wrong. The risk of major global catastrophe will be higher if everyone acquires nuke. The fact remains, major powers have nuke to prevent each other from mutual destruction and destruction to earth as a whole. Major powers do not need nuclear weapon to defeat smaller powers. They can do so conventionally. Now if small states have nuke, the possibility of using nuclear weapon will be exponentially higher and that is dangerous to the world. Therefore, NPT is design to block that movement. Big, medium, and small states all agree on that.
Major power can use their military might to bomb other nation without a justifiable cause then small nation in the possession of nuclear weapons can use nuclear threat to protect themselve from a warmonger powerful nation.