Wouldn't that also disable other features as well (push notifications, maybe?)? Animations are something I disable permanently just to have a better user experience, I wouldn't like to sacrifice anything else.

Yes it would. But you can only reduce effects:

Settings -> General -> Accessibility -> Reduce Motion.

The only thing I dislike is the slightly counter-intuitive quick fading effect when minimizing or switching between apps. Outside of that though, any iDevice feels snappier with that option enabled (== effects reduced).