Why would you cleave your chicken, bones, into pieces when you are making stock? I just throw whats left of my chicken in a pot with some vegetables bring it to a boil and then simmer away for an hour, am I missing something?
To release the bone marrow, which is excellent source of protein and has medicinal properties.
The collagen released from the chicken bone marrow is what gives the soup body; It's why the soup feels more substantial in your mouth than a spoonful of water. As the stock cooks, the collagen breaks down into gelatin, if your not seeing this, you may need to cook the stock for longer.
Obviously if you don't break the bones, the marrow cannot be released. That having been said, a large amount of fat is also released from the marrow which later has to be separated from the finished stock. I have made perfectly excellent soup without bones, in fact Cooks Illustrated suggested using ground chicken which contains no bones or marrow (albeit with some prepared stock).
I suggest you find a traditional stock recipe and follow it's instructions. You may find that you've been missing something, or that you like your current recipe more.