"The Church received from the Apostles the tradition of giving Baptism even to infants. For the Apostles, to whom were committed the secrets of divine mysteries, knew that there is in everyone the innate stains of sin, which must be washed away through water and the Spirit"and St. Cyprian
"In respect to the case of infants, which you say ought not to be baptised within the second or third day after birth, and that hte law of ncient circumcision be regarded, so that you think that one who is just born should not be baptised and sanctified within the eighth day,we all thought very differently in our council. For in this course which you thought was to be taken, no one agreed; but we all rather judge that the mercy and grace of God is not to be refused to any one born of man.. we ought to shrink from hindering an infant, who being lately born, has not sinned, except in that, being born after the flesh accrding to Adam, he has ontracted teh contagion of the ancient death as its earliest birth, who approaches the more easily on this very account to the reception of the forgiveness of sins -- that to him are remitted, not his own sincs, but the sins of another (Adam)"So scripturally, traditionally and historically there is no case to deny infants baptism. On the contrary, one finds scriptural proof of baptism as the New Testament's circumcision where entire households are baptised and one finds historical proof of it being practised by the early Christians.
My point on SS: even if there was no church history or recounting of infant baptism from the period, Scriptures itself speak through the words: all/paedia/household etc about the universality of the practice and the lack of exclusionary words. When Jesus speaks of all men he means all people not just males.
Regarding symbol v. sacrament. Baptism has God’s Word and an external element. It is a work of God for us, not a work of man.