Factually incorrect. Infant baptism was practiced long before 325 (what an interesting date to choose). More on that later.
I suspect that your interpretation is greatly infulenced by a hyper individualistic culture. Of course, when an adult is baptized he or she should repent and be becoming Christian, and if a child cannot speak, let the adults speak for him. When both parents are Christian, it is assumed the child will be too. They baptized whole households. Baptism is the new circumcision; an adult convert obviously needed to believe in Judaism, but a child of Jews was expected to be circumcised as an infant.
Infant baptism in the early Church is attested to here:
He [Jesus] came to save all through himself; all, I say, who through him are reborn in God: infants, and children, and youths, and old men. Therefore he passed through every age, becoming an infant for infants, sanctifying infants; a child for children, sanctifying those who are of that age . . . [so that] he might be the perfect teacher in all things, perfect not only in respect to the setting forth of truth, perfect also in respect to relative age” -Ireneas, Against Heresies 2:22:4, written around 189. Notice who is to be reborn here.
“Baptize first the children, and if they can speak for themselves let them do so. Otherwise, let their parents or other relatives speak for them.” -Hippolytus, The Apostolic Tradition 21:16, around 215
"Every soul that is born into flesh is soiled by the filth of wickedness and sin. . . . In the Church, baptism is given for the remission of sins, and, according to the usage of the Church, baptism is given even to infants. If there were nothing in infants which required the remission of sins and nothing in them pertinent to forgiveness, the grace of baptism would seem superfluous” -Origen, Homilies on Leviticus 8:3, around 248
“The Church received from the apostles the tradition of giving baptism even to infants. The apostles, to whom were committed the secrets of the divine sacraments, knew there are in everyone innate strains of [original] sin, which must be washed away through water and the Spirit” -Origen, Commentaries on Romans 5:9, around 248
"Then people brought little children to Jesus for him to place his hands on them and pray for them. But the disciples rebuked them.
Jesus said, ‘Let the little children come to me, and do not hinder them, for the kingdom of heaven belongs to such as these.’"