I believe it absolutely would be an issue. While Sagan may have exaggerated on it's severity however fallout is still extremely deadly. Being exposed and downwind of a single thermonuclear warhead could give someone a lethal dose inside of an hour even a hundred + miles away (depending on wind of course). There would also be significant genetic damage done to those who do survive making healthy offspring far less common. Imagine what over a thousand warheads detonating in North America would do.

Some estimates also have up to 80% or more of the ozone layer being destroyed in the norther hemisphere which would drastically increase UV exposure likely killing huge swathes of essential microbial life. It's not about everything turning green and melting but rather a cascading collapse of the world's ecosystems creating an ELE not seen in tens of millions of years. Nuclear winter would of course be bad, extremely fucking bad but the amount of ejecta/black soot would be significantly less than a super volcano like TOBA and roughly 2-5x that of Krakatoa depending on the altitude the bombs are used at.

In conclusion, as a total non expert... I don't believe it would wipe out life but It could make much of the biological world we know today unrecognizable in many ways and human civilization and the world around us would absolutely be brought to its knees.