On small screens....
" On small screens, a flatter UI can have some advantages because it uses less screen space (less relevant today because of high res high dpi screens). "
Actually, on small screens, 3D-effect controls are even more useful, because most small screens are now touchscreens.
One of the less-discussed benefits of the 3D button is the matter of target areas. To the user, the click target is the "top" of the button, or the inside of the check-box or radio-button. To the computer, the whole area is a valid target area.
This means that the whole area containing 3d effects becomes part of the margin for error in user clicks, and allows you to put buttons closer together without compromising the usability of the interface. This density of buttons is often visually more appealling than using blank space as padding between buttons.
When you're interacting with your finger rather than a mouse, you need a much wider margin of error, and in flat interfaces this often leads to sparseness of elements, and only the centre of the active area is marked -- the boundaries are absent. This makes it significantly easier to accidentally go over the boundary and hit the next interactive element.
For example -- the YouTube interface. Even on a medium-small device like the iPad Mini, it's not uncommon for me to accidentally press the timeline and rewind a video instead of playing/pausing it. This is pretty frustrating.