Max range for birdshot is ~120 feet so a drone can just increase its altitude and be safe. Also you have to be aware of what is beyond your target which might not be possible in an urban environment.
Effective range for taking out a drone is "as far as the bird shot will shoot" which is much further than ~120ft. We aren't talking about effective range (e.g "breaking a clay plate").
That being said - your point about increasing altitude still stands. I'm not sure how birdshot will go when shot near vertically - but it can travel well over 200 yards horizontally depending on if you're using 04 or 08, etc. At a certain point its harmless to humans unless it hits in the eye but as long as it hits a blade of a drone it could still damage the drone and bring it down. It doesn't need to completely destroy the drone like a point blank blast of a double barrel with buckshot would.