Why should it sacrifice itself? There are more than fast-flying kamikaze drones. Smaller spotters or all those improvised models based on commercial quadrocopters with an AT warhead are nice targets for a properly sized bird of prey coming at them like they would hit their regular prey... fast and from top.
In fact there were several project world-wide to actually train birds to hunt down drones. Most were abandoned as fast spinning rotors can be a danger to claws when they get unlucky... but still it's perfectly reasonable to expect a big owl, eagle etc. to down a drone.
yes actually. I saw a video of training for this long before the war started. Surprised nobody has tried it yet, I assume it's probably pretty difficult to fight actual birds with controllers and a ton of delay