New York is where both engines of USAir flight were hit by soft bodied geese weighing less than 20 pounds each and forced the plane to crash land in the Hudson river. The drones have hard metal parts and hard plastic. They would do far more damage to the plane.
No, a drone would probably not damage a plane in the manner of the 'Miracle on the Hudson'. Even if a drone took out a single engine, all planes and pilots are certified to fly on the remaining powerplant. The problem is that the US Air plane ran into a flock of geese which took out both engines simultaneously. At least so far, drones have not been flocking (that would be scary.
I do think that one answer to this is to develop small, low power transponders that will fit on a drone. Should be possible and then should be absolutely required for any drone over a certain size. If you can see them, you can avoid them. If they are serialized then you can go after the miscreant without a whole lot of fuss and bother.
Although a little heavy handed, you could rig a system where larger drones won't fly unless they have the transponder and the transponder is registered. Yeah, somebody would hack around it but most people wouldn't care and in fact you could use the transponder to find the thing.