Do you know that it actually fires bb's out in a cone shape? If you aim a shotgun up in the air, you are not taking out any drones.
Look up the video of the drone hitting the hotel in Bahrain to get an idea of the speed and altitude.
Both sides have been seen with one member of a squad carrying around an issued shotgun in an anti-drone role- the fact that it shoots pellets in a cone is precisely why it's so effective. Skeet shooting is a great example of how relatively small fast moving targets can be hit consistently at range with a shotgun and they are usually using much smaller/lighter pellets with poorer velocity/range, I would assume the loads used in an anti-drone role are bigger.