Russians want the world to believe that they can do anything - this is not true.
Russian nuclear forces are composed of different types of ballistic missiles. Many are old designs and not accurate. CEP of relatively new RS-24 (Yars) is poor as well. Russian nuclear force is suitable for strategic military applications - to wipe out cities and other clearly visible targets. Russians cannot wipe out well-prepared American military installations, however.
American nuclear forces are optimized for tactical military applications on the other hand - to wipe out anything of value including military installations and deeply-buried facilities.
The combination of a lack of Russian situational awareness, dangerously short warning times, high-readiness alert postures, and the increasing US strike capacity resulting from a new fuzing system for submarine-based nuclear missiles has created a deeply destabilizing and dangerous strategic...
thebulletin.org
By Hans M. Kristensen Under the cover of an otherwise legitimate life-extension of the W76 warhead, the Navy has quietly added a new super-fuze to the warhead that dramatically increases the ability of the Navy to destroy hard targets in Russia and other adversaries. In a new article in the...
fas.org
There are also classified American military technologies to factor-in. How many Russian ICBMs can they disable without firing a shot? One American military official hinted something to this effect recently.
Your assumption is based on Nuclear Winter hypothesis. Russia will be able to radiate some locations but not every location on Earth in a nuclear war. What you see in Terminator movies is the most likely outcome
minus the machines trying to kill every human.