Pretty much this is true, if internet goes down bitcoin network will collapse (we are talking about the global internet shutdown here). Furthermore in transaction like payment online or remitances, as stated, when bitcoin network collapse all the transaction will be at halt. So no remittances, even all sort of online transaction will stop.
Intercontinental cables are the most vulnerable pieces. Intracontinental ones, on the other hand, have a good chance of surviving. Bitcoin transactions can still be broadcast, but it won't reach the other side of the world. Miners can still receive transactions, but only from the extent of their connection (e.g. a US miner can't receive Chinese transactions since trans-Pacific cables are offline). There would be a fork basically.