True, it can be monitored, and it is, but there are good ways of encrypting your communication if you are worried about this (SSL, PGP etc).
The Internet is based on the TCP\IP protocol, and in essense this is a distributed network. This means there is no central server controlling it, and hence you can't shut down the internet with a few switches as you say. It was developed specifically to be able to provide a working communication network AFTER a nuclear attack. The idea here is that taking out a server here or there doesn't really affect the network as the data will just be routed through whatever servers are left.
Seeing how anyone anywhere in the world can set up their own server, there is no way to control it or shut it down. The idea that the US controls the Internet is plain silly. As others pointed out, why are there then Jihad websites being allowed to operate? Because the US can not go to the country where the server hosting that site is located and remove it...
The only real weakness of the Internet is the DNS "backbone". This is a collection of servers that remember domain names (such as abovetopsecret.com) and connect them with the IP address of the server hosting the site (in this case 75.126.76.151). If these servers were all shut down you wouldn't be able to access any websites without knowing their IP address. Since most people don't go around remembering IP addresses, DNS is a vital service. But even with it removed, the servers and all their content is still there! It's just trickier getting at it. And hackers have tried shutting doen these servers with DDOS attacks, but it didn't have any real effect. They were able to slow down or stop a few of them for a few hours, but there was plenty of redundancy left to keep the Internet working just fine.
So in short, no the US can't control the Internet. They actually designed it that way... They can however deny you access to it if they controll your ISP, but then again there are lots of ISPs to choose from and lots of different ways to connect, so in reality this kind of control is impossible.
Edit:
The vid talks about the US controlling IP allocation, and is in a way true. ICANN does this job, and I guess it's a US company, but wikipedia lists it as a non-profit corporation. Maybe somebody else here have some dirt on them?
"Prior to the establishment of ICANN for this purpose, IANA was administered primarily by Jon Postel at the Information Sciences Institute at the University of Southern California, under a contract USC/ISI had with the United States Department of Defense, until ICANN was made to assume the responsibility under a United States Department of Commerce contract."
http://en.wikipedia.org/wiki/Internet_Assigned_Numbers_Authority
http://en.wikipedia.org/wiki/ICANN
So if anything is controlling the Internet is commercial interests I guess=)