I've gone to church all my life, and have had my full share of experiences with screaming children in church. It gets really bad at Christmas and Easter, when many families make their "annual trek" to church, with children who have no clue as to what behavior is appropriate.
Familes that bring their children every Sunday, all year, usually have well-behaved children. The children are familiar with church, they know what to expect and they are under peer-pressure, as well as parent-pressure, to act in age-appropriate behavior. No one wants to go to Sunday School and hear "Hey Tantrum Baby!" from the other kids.
How to address this issue?
1. Yes, removing children from church will just serve to reward the child for bad behavior. I did that, myself, several times with one of my twins. My approach was to make the time out of church unpleasant. I would take my child to an empty room, devoid of toys or other distractions, then make him sit facing the blank wall. He could not climb in my lap. He had to face the wall and calm down. Then, when he was ready to go back into church, we'd go.
2. Churches should have ushers who guide the parents of really unruly children to such a place where the child's needs can be met. The hungry child can be fed, or even breastfed in private. The sleepy child can be put down for a nap. The poorly-behaiving child (such as one of mine) can get a time-out. The over-stimulated child can be calmed down.
When parents are unfamiliar with the church (such as the above-mention C/E only folks or when attending a church service away from home), they don't know where they can go with their kid. That's the job of ushers to guide the parents to a safe and private place for the parents and kid.
3. If a child near you is making so much noise that you cannot hear, then you have several options. Turn around and tell the parents "I'm having a difficult time hearing. Could you ask Sonny to be quiet?". If that doesn't work, then just get up and move to a better seat. If you cannot get up, then raise your hand and say "I'm having a very difficult time hearing you. Is there any way the sound system can be increased in volume?".