I'd be uncomfortable emphatically staking a position on the whether a global government would go bad. There are many factors aside from ideology that determine if a government functions well; if someone tells me "I'm forming a new liberal democracy" it is too little information to predict if it will be a good one, or a bad one.
People do overestimate the risk that a nuclear war would lead to human extinction. Unfortunately, people also overestimate, gravely, the capacity of humanity to refrain from nuclear war. It has been less than a century since Hiroshima and Nagasaki, and we've come close to a nuclear conflict on several occasions.