I noticed that Elder Kraken seems to have the hardest time to destroy a shield drone fast (besides it being genrally too strong). His air burst does not reach that far, to hit it with 3 melees right now you may acutally use 2 burst to get there. The best tool to destroy it would be the BM, but thats his best burst damage ability actually.
If you dont want to waste too much you therefore would use Lightning Strike.
Goliaths have Flamebreath which does not need to be applied for the full duration to destroy the SD.
Kraken has snowballs and Banshee Mines. Using 1 Banshee Mine may only be a third of an ability.
Wraith only has Warpblast actually, but the traversal is fast and good to get into melee range.
Behemoth may use a Lavabomb, which prevents another shield drone there for some time, or Rock wall.
Gorgon best uses Acid spit, where one pool on the SD is already enough to destroy it.