There's a fine line between a "sport" and a "game". A game, to me, is played purely just to see who's best. It's played just for bragging rights on who beat who.
A sport, to me, is where a challenge is presented and we see if we can overcome it. If we can, fine; if we can't, then we continue onto the next one. With nothing lost, and ego untainted.
You speak the truth in your post. It's the whole experience of the chase, and every animal is a trophy.