To show, you want everything you do to cue your horse to be almost invisible. On a well trained western horse, a flick of a pinkie finger can send them into a spin - This is because the curb bit magnifies a small signal on the rein to a larger signal in the mouth. So an invisible movement of the hand becomes a signal the horse can easily decipher and respond to in the mouth.
You can certainly do the same things in a halter or sidepull - But you can't do them with the flick of a finger. You need to open reins and move your whole hand. Do you see what I mean? The bit means the rider has to do less to acheive the same or better result.
That's all we mean by 'refining'.
Riding bitless is great of it works for you. Many horses are indeed happier that way. However performance horses generally DO need a bit. We need signals to be invisible, response to be immediate.
I plan on eventually riding my Arab mare bitless for endurance. It doesn't matter how big or messy my cues need to be out on the trail.
However when I show her or do any speed events, I will be using a bit, because then I need the more precise communication and refined cues.
Do you see what I mean?