I completely understand your point of view and, some time ago, I was thinking exactly the same. However, pain is a defense mechanism needed to protect your body also from yourself. I agree that, most of the time, there isn't much we can do about it, but you should also see it the other way round: a broken finger that hurts won't be used. If pain subsides, we naturally tend to forget about the injury and, without thinking, we use the damaged body part, worsening the condition.
I have a perfect example of this. Many years ago, a classmate's grandma started losing sensitivity in her left hand. She didn't mind it much, to her it simply meant that she couldn't feel much with it, including pain. One day, she inadvertently put the hand on a hot hob, and she noticed only when she smelled it. To her it felt like mildly warm. They had to rush her to the hospital with severe burns, which she didn't feel, and tie her arm so that she wouldn't use her hand while it was healing.
Simply put, pain can save us from ourselves (although I agree it can be so annoying).