When it comes to teeth pulling, it's best to let the kids do it themselves.
Have the child wiggle it back and forth with clean fingers or their tongue until it comes out - if it's really loose, it won't hurt and there will be minimal bleeding. If there is a significant amount of bleeding, you can hold pressure with a wet gauze to the gum for a few moments to stop the bleeding. My concern about an adult pulling a child's tooth is that the adult doesn't know how attached the tooth is or how much it hurts. However, it sounds as though your child's tooth was ready.
By the way, a swallowed tooth is not harmful, just a little disappointing to the tooth fairy.
Legal disclaimer: I am providing this general and basic information as a public service and my response to this question does not constitute a doctor-patient relationship. For any additional information, advice, or specific concerns, please speak with your own physician. The information provided is current as of the date of the answer entry.