Where the problem arises in my own experience is with your [or others] definition of a 'public place'. A movie theatre, for instance, is a public place. If your child's screaming interferes with my own ability to hear the movie, then I think it is well within my rights to complain. If you don't want me to complain to you, then I will complain to the management. Which would you prefer? I don't care either way, I just want to hear the movie.
Church is also a public place and I am very tolerant of children's need to make some noise, so parents who give children toys and snacks in the pews don't bother me at all. However, if you allow your child to crawl under the seat and poke at the toes peeping through my sandals, then I will object. I promise not to kick your child, but I am likely to bend down, stare him/her straight in the eye and say, "I am listening to Father right now. He is talking about God. I find it hard to listen to him when you are poking me. Please go poke Mommy or Daddy instead. Thank you." [Children seem to respond to this by quietly crawling away FYI.]
Another public place is the school system. If my child is attempting to learn math facts, then I have to admit that your child's screaming or running is likely to interfere with that. Since the purpose of school is to educate children in the approved curriculum, I would expect that someone is going to interfere with your child's disruptive behaviour.
That's life, you know? I don't get to run screaming through public places as an adult, and somewhere along the line children have to learn that they can't either. If you are teaching them that, I applaud you. If you aren't, what's stopping you?