I can't see why the internet would reset us to the the bronze age. I think what they're saying is it's more likely to reset us to similar state to when the internet wasn't common-place, ie, the 80s.
I should've clarified - my memory is that even in the 80s the 'internet' wasn't that fundamentally-critical to typical daily behaviour. Heck, even the WWW took a while to roll into relevance.
Not a chance, the changes wrought by the industrial revolution would not be undone so easily. Removing the internet from society would not change what people already know how to do and what is possible to do; other methods of exchanging that knowledge (schools/books/etc) would be rapidly re-emphasized. It might even cause a reversal of the compency crisis as people wouldn't have the rest of the internet to distract them from their studies...
Realistically however, attacking some undersea cables isn't a threat to the existence of the internet anyway. Even if you could split THE internet into many national/regional networks, that would still be sufficient for most industrial and economic needs. More expensive and robust links could connect these regional networks where truly needed.
The US is special in being able to build and sustain huge tech companies. But the most important (internet or otherwise) services for anyones' economy don't require such huge companies.
Not saying that the loss worldwide free communication wouldn't be painful, but I think it would be as painful for both the providers and the consumers of services.