As with any family in America, I'm sure there are positives and negatives in how they are raising their family. I know they are part of the quiverfull movement where it is implied the more children you have, the more godly and obedient you are.
The children are well mannered and well taken care of financially. The father and mother seem like sincere, good parents wanting the best for their children. Unfortunately, the number game plays into this. When you have a sign up sheet on your refrigerator where you can have an hour of one on one time blocked off each week to spend with either mom or dad, that's a problem, not to mention the almost playoff seeding format in having the older children take care of the younger ones (not assist the parents...they are "in charge"). Also, the fact the family "is" their church raises red flags, and that they have turned their children into actors.
God has given us the ability to reason and use common sense. He has given us knowledge in regards to reproduction. We KNOW that when a man has relations with a woman on certain days during the woman's cycle, barring any problems, the woman will get pregnant. This is God's design. If you have s*x during that time, God has designed you will get pregnant. Having relations during this time is not trusting God for the size of your family, it is testing Him. It's like walking across a freeway during rush hour and telling God "if you don't want me to get hit by a car, I won't...I'm trusting you".
Using birth control does not trump God's sovereignty. If He wanted you to have a child, no manner of birth control will prevent it. To plan every aspect of our lives but this one is irresponsible, IMO, particularly if the children's needs cannot be met (financial, physical AND emotional).