It affects other species as well. Also, eagles becoming extinct will have far more consequences than a few fewer birds in the sky. Since they are apex predators, everything below them in the food chain will be affected. And guess what's also part of that chain? Farms and livestock.

Ecosystems are complex enough that we have very little information on all the interactions that could happen. It's more than just a bag of organisms mixed together. A better analogy is the network of files and components loaded up during the boot process. Especially early on, each piece is crucial to what happens next.