I mean this theoretically works until the gangs decide to start putting small bombs in the drones that detonate on impact. I bet they can build drones faster than people can breed and train the birds.

Well yeah then you have a counter swarm of exploding drones. The government can always afford more than the criminals because they can just squeeze their tax cattle a little harder.

It doesn't just theoretically work.

Trained falcons and eagles are being used by police forces already:

https://www.google.dk/search?q=training+birds+to+attack+dron...

I don't know if any counter-measures are getting popular yet, but it's probably cheaper to poison or shoot them than explode them.