Nah, modern bomb yields are significantly lower. The actual innovation is in something called MIRV, which simply put is an ICBM that splits into many smaller bombs and decoys. Modern warheads are also dramatically more efficient than the early ones you're talking about, there would be much less fallout.
Nuclear war is still a tremendously bad idea, just not because we'd be dropping humongous weapons on each other that then irradiate the atmosphere for centuries to come. Actual fallout from modern warheads is really very low. Humanity would live on, nations wouldn't. It's just that a world without Russia wouldn't be worth living in.