Of course you're correct. Here's a 2019 simulation of escalating full-scale war between the US and Russia, done at Princeton based on what's publicly known about actual force postures. Immediate effects would be 34 million dead, 91 million total casualties, and 30 major economic centers destroyed on each side.
https://www.icanw.org/new_study_on_us_russia_nuclear_war
Fallout would cause significantly more casualties. Then there are the environmental effects. Science on that has evolved over the past few decades, so wikipedia is a good starting point for an overview.
https://en.wikipedia.org/wiki/Nuclear_winter
Since none of this is certain and we're talking about risks, considering the worst cases seems appropriate, and those involve agricultural collapse, several billion dead, mass extinction, and an ozone hole covering most of the planet. It might not kill off the human species, but modern civilization would struggle to recover.