Many parents can not afford to send thier children to daycare or don't have any options available, and end up staying home from work or quiting their jobs. There needs to be a better system in place, and the one proposed by Germany could be the catlalyst for some major changes.
Yes, the U.S. should adopt a similar proposal because all parents should be able to work while their child is in daycare. If a parent cannot afford daycare, a city or state should given them daycare for free. If they have to stop working instead, it will keep the family poor.
A lack of affordable child care facilities make it almost impossible for women with young children to enter the workforce, contribute to their family's income and develop their own economic independence. The problems caused by a lack of affordable child care fall disproportionately on the poor and single parent families meaning that inequality is exacerbated. This is true not only in Germany but also in the United States and women should have compensation.
People that want to become parents need to figure out in advance how they are going to support the basic needs of a child. If they can't do it, then don't have them. Your child, your responsibility. Not the gov't or other taxpayers. You need to care, feed, provide shelter and healthcare to your child(ren).
The German rule that parents without day care options can sue cities for lost earnings is one that should not be applied in the U.S. Parents should shoulder responsibility for the care of their own children. If they can't find day care options, it is up to them to solve the problem, not the city. Personal responsibility is an important feature of American life and philosophy.