Perhaps. Perhaps not. But it will survive it. It will survive a complete nuclear winter. It's too useful to die, and will be one the first things to be fixed after global annihilation.

But Internet is not hosting companies or cloud providers. Internet does not care if they don't build their systems resilient enough and let the SPOFs creep up. Internet does it's thing and the packets keep flowing. Maybe BGP and DNS could use some additional armoring but there are ways around both of them in case of actual emergency.