ORIGINAL: jester_s1
If we want to be technical about it we need the up elevator to maintain altitude because when we bank the plane we are using some of our lift for turning. Since a portion of the lift we were using to fly straight and level is now being used to push the plane to one side, there isn't enough anymore to maintain altitude. So turning requires the wing to make more overall lift, thus the change in how much elevator is required to maintain a constant altitude.
That's true. But it's also true that up elevator is needed to turn when banked. It does two useful things in turns, not just one. As you said, the banked wing "pushes" the plane sideways. Some up elevator helps convert that push to a turn. Trying to tighten a turn by using just more aileron doesn't work, and not just because it loses altitude. This is why Langewiesche disliked the term "elevator." He called them "flippers," which didn't catch on.