Breastfeeding mothers should not have to search for a place to breast feed if their child is hungry. You do not have to watch. If people can eat how they want when they want in public places a baby should be able to also. I would not want to eat in a restroom or with a blanket over my head and neither a baby. I think America has forgotten what breasts are actually for.
Yes breastfeeding is important, more to the child than the mother. While it does create a bond between mother and child breast milk is also the best choice for babies, if given from a healthy mother. America is so sensitive to public breastfeeding because they aren't educated enough on the subject to not find it uncomfortable.
No, breastfeeding in public is not important to a mother's health, because there are plenty of ways for a mother to be discreet. A mother can find a private room to go to where other people do not have to witness it. Those who want to breastfeed are welcome to do so, but that does not mean that the rest of us have to watch.