It should, but they have to be under user control at the time you destroy it. I always let them do their own thing. Never found a PUG that is gonna watch your back while you work towards that next kill streak. Heck in HC they will shoot any scorestreak they see, enemy or friendly.
I just finished this. It took awhile. I always run with a laucher because I like to shoot down uav/cuav and stuff like that. Sometimes that +75 pts from a blackout is all I need to get my next scorestreak.
I got my last one from shooting down this guys lodestar. Nothing like a good ol uppercut. lol
Grief, your wish may be the game's command... after a double XP weekend I am sure a lot of people will be prestiging and forced to play with the default scoretreaks for a bit, including the Dragonfire.