If any interest exists I can refer certain works for consideration over these and other vague terms. To continue to debate over what is and is not sport by brief refutation and assertion on the part of many writers is indeed interesting but the complexity of the question is often underestimated and under analysed by such brief rejoinders. The reason athletes get hurt is quite simple, you are paid to win. Pain is the currency. While I was raised watching and at times attempting to play American football, you have to have the ability to give and take pain.
Cycling certainly has that.
For example, a defensive lineman's job is to crush the guy with the ball. In fact, the ball carrier is supposed to go home and have nightmares about going on your side of the field. That's called fear and respect. You get these qualities by inflicting biblical amounts of pain on your opponent. In order to have this level of fitness, you have to train very, very hard.