Your connection is not going down and independent sites will stay up. Taking down everything would mean someone breaking all the local ISPs where you live, because otherwise you could just connect to your neighbors or public wifi. And most governments would do everything they can to fix it ASAP.
The only realistic way to do that is somehow finding a persistent DoS or RCE for every router vendor, hacking into an IXPs control systems, or physically damaging equipment. AI can't do any of that alone and I dont think it ever will.