You're right! That does seem like a lot of risk to be taken on blithely.
If I may offer a few points, in the interests of helping you understand why people might make a decision that seems so obviously silly?
First, bandwidth is still fairly expensive. It adds up much faster than many people expect. A CDN, especially one that doesn't meter or charge for bandwidth, is a lifesaver. Yes, it's adding a complex potential point of failure to your system. As you completely correctly point out, that is not a desirable thing. However, what you get in exchange for that complexity and potential failure is more controllable costs and a greater ability to fend off some major attacks.
On the control point, you're once again absolutely right! With the potential caveat that this puts CloudFlare on a level with every single other service on the internet. Unless you're one of the few people who owns all the infrastructure starting with your server and ending with a full backbone network, you're already relying on capricious companies vulnerable to legal coercion. So this isn't really a new risk for most people. It is, again, a risk people take because they judge the costs and added risks to be worth taking on.
And one would have to work pretty hard at it to get a fake DMCA notice through. CF's legal department doesn't screw around.
So you're absolutely, completely, 100% right on all counts. It's a lot of risk to take on, and your concern is wise and justified! For a lot of people, the gains to be realized justify the risks taken.