Main difference between their view and yours seem to be that they see God actually giving saving faith to the baby!
No, it is ALWAYS God that confers saving grace. Lutherans are in complete agreement with Catholics on infant baptism. You see, parents bring their babies to the waters of baptism by professing a belief in Christ on behalf of the child, and promising to raise him or her in the faith. For adults who are to be baptized, the Church also requires them to profess their faith in Christ.
Because baptism confers saving grace, the earlier a person comes to baptism, the better.
In infant baptism, then, though the child is too young to have faith, the parents extend their faith on the child’s behalf.
On what basis does the Church believe that the faith of one person may be effective on behalf of someone else? The Scriptures are full of examples in which Jesus extends healing grace to people based on the faith of others.
For example, Jesus forgives the sins of the paralytic based on the faith of those who brought him (see Mt 9:2; Mk 2:3-5).Jesus heals the centurion’s servant based on the faith of the centurion (Mt 8:5-13).Jesus exorcises the child’s unclean spirit based on the father’s faith (Mk 9:22-25).
We might also note that in the Old Testament, God spares the firstborn child’s life during the Passover based on the parent’s faith (see Ex 12:24-28).
Given these examples, then, we must ask ourselves: If God is willing to effect spiritual and physical cures for children based upon the faith of their parents, how much more will He give the grace of baptism to children based upon the faith of their parents?