Here's a perspective for you... What Christ did on the Cross was the culmination of the Marriage Feast of the Lamb begun during the Last Supper. In this celebrated liturgy, we were brought into the family of God as Christ's bride, the Church. The ritual and realities conform to ancient convenant ceremonies where families were formed. It is more than pardon, it is adoption.
In Baptism, we are washed clean and we take on God's Name as a new creation (we are Baptized in the Name of the Father, Son and Holy Spirit). Just as the neighbor kid can't become a member of my household on his own, we couldn't become members of God's family without Him covenanting with us. There is no reason that the infant that Christ said should "come to" Him, can't be born again into this reality of God's family just as the infant was to be circumcised in the old convenant to be a member of God's Holy People, Israel.
Perhaps not formally - I think it stands as a Mystery.
Archbishop Sheen compared original sin to a stream originating from a polluted place, with all of the water carrying the pollution of original sin.
As with other Mysteries, if you think you really understand it, then you don't.