The only difference between YouTube and Amazon EC2 is flexibility of purpose. You can do many different things with an EC2 account because it was designed to let you do those things. You can upload videos to a YouTube account and not much more because YT was designed to do one thing well.
What they have in common is that they are both hosting services and apps platforms. They both host your data and applications. The fact that the "window" into EC2 from the outside is whatever the account holder wants it to be, whereas the window into YT is just YT's interface... is totally irrelevant.
The bottom line is that both Amazon and YouTube are hosting your data on their servers, and allowing you to present that data to the public at large. If one of them is responsible for legality of your content, then both of them must be.
In the case of EC2, Amazon owns the building, and lets a space to you, to do with more or less what you will. If that happens to be running a bar that serves underage drinkers, how is that their responsibility?
Edited to add: Except, of course, in the case where they've been notified that you are, in fact, serving underage customers. Most leases have verbiage that allows the lessor to break the agreement and boot your ass if you're using the space to break the law. That's how they qualify for safe harbor.