Oh, I agree. (I assume you mean "fusion" rather than "fission", but that doesn't really matter).
The idea that all life or even all humankind would be destroyed is fanciful.
I think it is true, however, that a nuclear war that involved, let's say, 70% of large cities would end life as we know it. All the structures of civilization - government, law, communications, transportation, material logistics - would be finished, and survivors would be scrambling for themselves for quite a while until some kind of civilization could be rebuilt. That will be a very nasty experience for all but the strongest, and possibly even for them. I would expect that some kind of civilization would emerge quite a bit faster than it did the first time around for us, because there would be people with memory of the technologies and institutions we've used, at least for a few decades. A lot would depend on how much they could pass on before dying off, because much of the information needed would evaporate from electronic storage. Books would become very important again.