If it actually becomes a problem, I start carrying my shotgun everywhere, but... Even in the US, most Americans aren't particularly good with a shotgun. Enough of us that most places could have a number of people that they just call. Wake up right now and get out because you might need to shoot something, but it's a dangerous job and I really hope that it never comes down to that.
I'm reasonably good with clay targets. I would expect that if I was actually asked to take on drones I would need to do some practice on drones before I'd be good at them too. Drones are more expensive than clay targets but they're not particularly expensive and considering the situation I'd be willing to pay the price of a few drones to learn how to shoot them down down in a controlled setting where it's safe.
Shotguns have very short range: if I'm expected to be taking down a drone with a shotgun things are really bad at that point my life is alread in danger. if I hit I may save my life.
Remember you are only seeing a selection of the thousands of times this has happened. The drone operating side will release more of the ones where it didn't work.
It does work at least sometimes. If it didn't work at all soldiers would not bother carrying the shotguns.
It would be surprise me if it is really true. It is probably one of the easier type of gun to pick up in terms of skill and shoot accurately. Still, 'shotgun minute men' has a ring to it.