Just to add, Android's material design is full of shadows and 3D encoding, while still being flat. It's not a great design by any means, but it's proof that flat design does not imply on a 2D UI (despite its name).
The single largest problem is that when we took 3D buttons away, we got no icon for the idea that you some object is clickable. An icon does not need to be skeuomorphic, just unambiguous and easy to recognize, besides, it needs to exist to be useful. Flat design uses a high-contrast background to encode that, what is extremely ambiguous, but there's nothing prohibiting people from creating a better icon.
The single largest problem is that when we took 3D buttons away, we got no icon for the idea that you some object is clickable. An icon does not need to be skeuomorphic, just unambiguous and easy to recognize, besides, it needs to exist to be useful. Flat design uses a high-contrast background to encode that, what is extremely ambiguous, but there's nothing prohibiting people from creating a better icon.