Eh, in general, but he gets a few things wrong.

Change the salesman to a page request and the houses to servers and you’ve got Internet packet routing.

Nope. Conceptually, what makes Travelling Salesman hard, is that you have to return to the starting city. With internet routing, you have to "visit" all the servers, to find the shortest path, but you don't have to "return." Once you've found the destination, you're done. Dijkstra's algorithm is a perfect solution, and runs in a very small P (specifically, O(|E| + |V|log|V|), where |E| is the number of connections between servers, and |V| is the number of servers).

If you could figure out a better way to route data on the Internet, you could make stock trades faster than everyone else and make billions of dollars.

Even if he was right, which he's not, you still couldn't 'make stock trades faster.' The internet uses Dijkstra's algorithm at a low level which users have no control over. Even if you figured out a better algorithm, you couldn't make the internet run it. You'd have to build your own internet.

There’s no good way to write a computer program to find the factors of large numbers quickly.

Actually, there are a number of special-purpose polynomial time factoring algorithms. If you have a quantum computer, there's even a good general-purpose one. There's no general purpose one for regular computers yet; but nobody really cared about factoring until 1970, so we've only been working on the problem for about forty years. There's no good reason to believe prime factorisation isn't P. Give it time.

If finding those factors were easy, then the Internet would fall over.

No, it wouldn't. There are numerous asymmetric algorithms besides factoring. When factoring is broken, we'll just switch to a different one.