Ryan you are correct. Baptism of babies and young children didn't come until much later. The reason for this is that a baptism is a dedication of yourself to God. How can you do that if you have no understanding of God or what he requires of us?
As this relates to children, even if God gave infants that grace how could they make the decision? How could they tell us they are ready?
Besides all this reasoning there is scripture to show that the child is protected through the parents. 1 Corinthians 7:14:
"For the unbelieving husband is sanctified in relation to his wife, and the unbelieving wife is sanctified in relation to the brother; otherwise, your children would be unclean, but now they are holy."
This is true even if only one parent is a believer. The child will be saved through their faith. God prepared for these circumstances ahead of time. He draws those to him, those having the heart condition and qualities God favors. Ephesians 2:8-10
"By this undeserved kindness you have been saved through faith, and this is not of your own doing; rather, it is God’s gift. 9 No, it is not a result of works, so that no one should have grounds for boasting. 10 We are God’s handiwork and were created in union with Christ Jesus for good works, which God determined in advance for us to walk in them.
As for the scriptures surrounding Prevenient Grace. I posted on a similar question with plenty of verse to support it. Prevenient Grace - Where/How is it Supported/Rejected in the Bible?