Headlines are almost
never written by article authors.
The corresponding chapter title from Smil's book is "Understanding Our Material World: The Four Pillars of Modern Civilization".
Smil highlights their significance, irreplaceability, and that "the mass-scale production of all of them depends heavily on the combustion of fossil fuels, and that some of these fuels also supply feedstocks..."
The title as submitted seems to me to be fair: "The modern world can't exist without fossil fuels". Not that civilisation cannot, or that there isn't a possibility for replacing either the materials to a substantial degree, or the role of fossil fuels in their production, but that the result of doing so would leave us with something much at odds with the present parameters of the indistrialised, technological, high-energy, fossil-fuel dependent world we take as a given.
And to that extent, Smil's argument seems quite sound. You'll find similar viewpoints expressed both by strong critics of the fossil-fuel world, say, Richard Heinberg, or by apologists for the fossil fuel industry, such as Daniel Yergin. To the extent that they all agree that where we are now is a direct consequence of human utilisation of fossil fuels, I find no basis for disagreement.
The question is what the future implications and prospects are. Fossil-fuel defenders see an unlimited future for that resource. Critics think we need to eliminate all possible use immediately. Smil, as he himself notes, tends to cut a middle path in his arguments, though that's really not the most valuable aspect he brings to the discussion. It's an absolutely unsparing assessment and quantification of fossil fuels' role and where that leaves us in considering future options.
It's a philosophy which earns him strange bedfellows and frequent criticism from across the growth-sustainability spectrum.
And I think you'll find that the more you rely on Smil's actual text, rather than third-party headline summaries, the more his argument and facts cohere.