To be fair in the novels its emphasised a LOT that the initial deaths caused by the asteroid strikes aren't the issue. Earth has a population of 13bn at that time and killing twenty, thirty, even fifty million people with asteroid strikes makes practically no difference to the planet (while being obviously a tragedy).

What kills billions is the fact food and logistics are completely disrupted and in many places there's no ability for the government to enforce law and order. Its people starving, dying of disease etc that ends up killing billions of people.