Now, to the issues.
If you're going to cite the ECFs you have to be prepared to accept all of their statements...not just the ones you like.
Augustine, like the other ECFs was all over the board on the issues near and dear to Roman Catholicism, so let's agree to reject him.
Second, we have to understand that baptism is not what saves you. If it were, the thief on the cross is not saved as he was not baptized. Further, some of your fellow RCs have admitted the actual act of baptism is not needed. The examples I use to illustrate this are a soldier mortally wounded on the battlefield. If he/she places their faith in Christ and are not baptized do they go to Heaven or Hell? Rome has to dance around this with a "baptism of desire" which is not attested to in the NT.
So No. Baptism does not save you. Faith in Christ, and only Christ saves you.
Now to the rest of the post.
The apostolic Church practiced the baptism of whole households, with no exceptions mentioned for small children (Acts 16:16, 33, 1 Cor. 1:16). There is no record anywhere in the Bible of a child of a Christian first having to reach the age of reason and then being baptized.
Whoa now....you've changed from infants, presuming these to be babies, now to small children.
Acts 16:16 should be Acts 16:15.
The Roman Catholic is making a HUGE presumption of the members of the households in the passages noted. In neither of the passages are any children mentioned nor infants.
There is no doubt that the early Church practiced infant baptism. Christ himself proclaimed the suitability of infants for initiation into the kingdom (Luke 18:15-16), and Peter declared: “Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit. For the promise is to you and to your children” (Acts 2:38–39).
15And they were bringing even their babies to Him so that He would touch them, but when the disciples saw it, they began rebuking them. 16But Jesus called for them, saying, “Permit the children to come to Me, and do not hinder them, for the kingdom of God belongs to such as these. 17“Truly I say to you, whoever does not receive the kingdom of God like a child will not enter it at all.” Luke 18:15-17 NASB
This is an interesting passage. It does not say the children were saved or healed after He touched them. It is possible they were sick....we don't know. When verses like this are vague in their meaning we have to rely upon the context of the overall passage and possibly the remainder of the NT.
Jesus is addressing a childlike faith to enter into the kingdom. Children are wonderful in that they simply believe what they are told. Their faith is a simple one in their parents...they trust their parents.
We should have such a simple trusting faith in Christ as well.
However, all throughout the NT it is clear one must have the ability to believe in Christ or reject Christ.
The Lord explicitly “called infants to him[self]” in Luke 18:15-17:
No....He said permit the children to come to Me.
When St. Paul led the Philippian jailer to Christ in Acts 16, he said to him, “Believe in the Lord Jesus, and you will be saved, you and your household” (Acts 16:31, emphasis added). He does not say that all in his household must first believe. He simply says they will all be saved. How could he say that? St. Paul seems to have understood what St. Peter had already preached back when Paul was still persecuting Christians (in Acts 2:38). The promise of faith and baptism is for the jailer and his children.
The requirement was for all to believe in the Lord Jesus Christ. If the jailer were to believe and his household would believe they would be saved.
This is why context is important in understanding the New Testament.
I cannot believe in Christ for the benefit of my family. If that is the case then all of my household, aunts, uncles, cousins, etc will be saved by my faith in Christ by Roman Catholic logic.
That is not attested to in the New Testament.
That is the most important aspect of understanding Scripture.
A Strong's and several good commentaries are essential to having a good foundation for understanding what the Word says.
I always find it amazing how tradition in the RCC tortures scripture to support non biblical theology such as Romans 5 :12