The world dismantled landlines, phone booths, mail order catalogues, fax machines, tens of millions of storefronts, government offices, and entire industries in favor of the Internet.
So at this point no, the world can most definitely not “just live without the Internet”. And emergency services aren’t the only important thing that exists to the extent that anything else can just be handwaved away.
There are likely emergency services dependent on Cloudflare at this point, so I’m only semi serious.