The term simple elbow dislocation may be a bit of an oxymoron (contradicting terms). As you said, there’s nothing simple about the elbow or dislocations. What the expression “simple” elbow dislocations means is that surgery was not needed to put the elbow back in place and there were no bone fractures. But with any trauma severe enough to dislocate a joint, there can be some soft tissue damage.
The elbow is made up of three bones: the humerus (the upper arm bone), the ulna (the larger bone of the forearm, on the opposite side of the thumb), and the radius (the smaller bone of the forearm on the same side as the thumb).
The elbow itself is essentially a hinge joint, meaning it bends and straightens like a hinge. But there is a second joint where the end of the radius (the radial head) meets the humerus. This joint is complicated because the radius has to rotate so that you can turn your hand palm up and palm down.
At the same time, it has to slide against the end of the humerus as the elbow bends and straightens. The joint is even more complex because the radius has to slide against the ulna as it rotates the wrist as well. As a result, the end of the radius at the elbow is shaped like a smooth knob with a cup at the end to fit on the end of the humerus. The edges are also smooth where it glides against the ulna.
There are also ligaments, tendons, muscles, cartilage, nerves, and of course, blood vessels in and around the elbow. Any of these structures can be stretched or otherwise damaged during the dislocation. Simple elbow dislocations heal well with few (if any) problems. Once the joint is reduced (put back in place), any residual problems may become apparent.
There may notice a slight loss of elbow motion, especially when trying to straighten the arm. There can be altered joint proprioception (sense of position), weakness and impaired motor control. Any of these problems can make sports participation or athletic activities difficult.