A drone caused issues with firefighters in California, and a drunk Secret Service Agent flew one onto the White House lawn! (Although I should point out that a man rode a full-sized gyrocopter onto the White House lawn as well, so let's not go blaming just drones.) There is a real danger that terrorists might use these explosive-delivery devices that is as terrifying as it would be simple to implement.
Look, none of us want to see these nice little devices accidentally causing harm. The few people who might want to use these with bad intentions - believe it or not - will not bother to register them. So, you have a bunch of well-meaning people registering their drones, and then they still send them into fire zones to get cool shots, but now you have someone to yell at for it. The bad people looking to blow up drones? They may never register them or they may mysteriously cut off the serial numbers before using them as floating weapons. Call me crazy, but I think there is a better way to solve this problem.
Like you have regulations around what frequencies the drones are allowed to use, it is reasonable to suggest a specific command on a specific frequency that all "hobbyist" drones must receive and obey the graceful landing command from. A small transmitter on fire engines, aircraft, around designated no-fly zones, etc. would be enough to stop drone enthusiasts' accidental wandering into the path of danger or causing undo concern. Right now, the FAA isn't arguing that anyone is doing any of this with malice - they are knuckleheads who just want to get a cool shot.
As for people with malicious intent, there are many intelligence services with overlapping areas of jurisdiction who regularly overstep their authority to spy on innocent people and are given a pass for it. If all of that loss of personal freedom and privacy isn't paying for their ability to stop bad people from doing bad things, then we have much bigger issues to deal with. Either way, the registry isn't the solution to this problem.
