Those beloved Church Fathers were quoting scripture just fine, and they interpreted them as the Apostles meant them to be understood.
The way Catholics today understand them.
Here, have some more.
“But the sacrament of baptism is undoubtedly the sacrament of regeneration: Wherefore, as the man who has never lived cannot die, and he who has never died cannot rise again, so he who has never been born cannot be born again. From which the conclusion arises, that no one who has not been born could possibly have been born again in his father. Born again, however, a man must be, after he has been born; because, ‘Except a man be born again, he cannot see the kingdom of God’ Even an infant, therefore, must be imbued with the sacrament of regeneration, lest without it his would be an unhappy exit out of this life; and this baptism is not administered except for the remission of sins. And so much does Christ show us in this very passage; for when asked, How could such things be? He reminded His questioner of what Moses did when he lifted up the serpent. Inasmuch, then, as infants are by the sacrament of baptism conformed to the death of Christ, it must be admitted that they are also freed from the serpent’s poisonous bite, unless we wilfully wander from the rule of the Christian faith. This bite, however, they did not receive in their own actual life, but in him on whom the wound was primarily inflicted.” Augustine, On Forgiveness of sin and baptism, 43:27 (A.D. 412).
“No sooner do they rise from the baptismal font, and by being born again and incorporated into our Lord and Saviour.” Jerome, Against the Pelagians, III:15 (A.D. 415).