Each time you kill a bird, he gains a buff. It stacks, and if it hits 5 stacks he will enrage. So yes, you do need to kill the birds, but you need to kill them quickly, before he summons more. He summons four at a time.
As Survival it's very simple. When he summons the four birds, just Multishot once or twice. They should die shortly before the next group. On my kill I didn't even need to switch targets... just Multi-shotted.
I did this one as BM. Kited him around a frost trap, pre-potted, and popped all cooldowns asap, readiness, rinse & repeat. Glaive Toss was enough to keep birds off of me and not enrage him too much. Recount showed me at 100k when it was over. This was a tough one for sure and took several frustrating attempts.