If Communism in its decline didn't bring about Nuclear War I doubt slightly worse working democracy would either. And on nearly any other kind of existential risk the difference between a poorly working democracy like that of Iraq and that of the US is probably very close to nill. Indeed the better working democracy of the US is far more likely to produce technology that increases existential risk.

Also arguably "democracies" work better with less voter input. No voter input translating into elected government didn't seem to hurt Belgium.