The FAA already has that authority.
Yes, you are right.14 CFR specifies that the FAA has authority over everything that is man-made and flies.
People are ignoring the rules, or just aren't aware of them. This not evidence that we need more regulations.
The FAA has authority to create rules, but the current set of rules need to be applied to newer technology. In short, the rules are limited to:
restricting operations to 400 feet above the surface; requiring that the devices give right of way to, and avoid flying near manned aircraft; and using observers to assist in operations;
What we should have is a set of rules which make a clear distinction between a "drone" and a toy aircraft. For example, I have one of those tiny Hubsan x4 quadcopters. The maximum distance it can fly is 300ft according to the spec, but by then I've already lost it as it is so tiny. A friend of mine has a $2000 GPS-equipped quadcopter with a call-home function. That would probably fit in the not-so-toy specification.
At this time, no skill-test is required to fly a heavy drone. All I'm advocating for is that we get people certified in rules and regulations, and make RC pilots aware of the NOTAM and TFR systems.