Actually, that's precisely the issue. If the entire internet goes down, it will start to impact things like the power grid and the water systems. Every year, utility companies put more and more of their infrastructure behind increasingly clever management infrastructure, and control it over the open internet. Hopefully none of it will fail-unsafe (though I bet something will), but if the management systems are offline for weeks?

And it will immediately, although not fatally, impact land line telephones, which I believe do significant backhaul over IP.