Not sure sterilisation lasts long on earth: life re-claims sterilised areas.
See your favourite massive volcano outbreak, or look at the Chernobyl exclusion zone.
(The impact of the disaster at Chernobyl is much smaller in area than a nuclear explosion would be, of course. But life has re-conquered everything there.)
In any case, I largely agree!
It might be worth noting that the number of direct deaths caused by Chernobyl is very low (2 perhaps?). But there have been hundreds of thousands of birth defects in its wake.
The survivors of nuclear war would have a really bad time, but humanity would survive.
None of this is to suggest we should be careless about it; it would be a massive disaster. But not the end.