There are two billion poor subsistence farmers largely disconnected from market economies who aren't dependent on modern technology so the likelihood of them getting completely wiped out is very remote.
The Earth land surface area is 149 million sq km and there are only about 12,500 nuclear weapons in the world. Even if they were 10 megatons each* and were all detonated, with a severe damage blast radius of 20km (~1250 sq km), it'd cover about a tenth of the land available.
Since the vast majority are designed to maximize the explosive yield, it wouldn't cause the kind of fallout clouds that people imagine, nor would it cause nuclear winter. It'd be brutal to live through (i.e. life expectancy of animals in Chernobyl is 30% lower) but nuclear weapons simple can't cause a human extinction. Not by a longshot.
* As far as I know, no one has any operational nuclear weapons over 2 megatons and the vast majority are in the 10s and 100s of kiloton range, so my back of the napkin math is guaranteed to be 10-100x too high.
Driving is a big problem for climate change as well but we can work on solving that one with EVs and a greener power grid. We don't have to permanently depend on fossil fuels to drive. But EVs still produce the same amount of microplastics that other cars do.
This is certainly possible, but if so extinction seems far more likely to come from nuclear warfare and/or disasters resulting from climate change. For example, it's become apparent that our food supply more brittle than many people. Losing access to fertilizers, running out of water, or disruptions in distribution netorks can lead to mass starvation by cutting the productivity we need to feed everyone.