Because there are multiple times the word "household" is used in Scripture. "Household" is a term for "family." And the assurance of salvation is made to families, over and over. It would be a real stretch to think there were no children in any of those households given assurance of their salvation.
I don't believe your skepticism over this fact precludes your children from being members of His covenant family. They will come to faith because they've been raised by a God-glorifying father and mother who may very well attribute their children's faith to their own good and righteous choice made after they are baptized.
But some of us will recognize that these children were given to your family by God for a specific purpose -- their salvation which will bring glory to God.
And you will not fail in that assignment from God because you believe in Christ as Lord, God and Savior. Which is what all parents need to keep uppermost in their minds and efforts (and which is confirmed in their children's baptisms) - that they have a profound responsibility to raise children who kneel to none but Christ.
Also, not everything Rome preaches is wrong (some days it just feels that way.)
Rome (ostensibly) is Trinitarian. Rome baptizes children.
But as with so much in Rome, it takes a valid instruction from God and turns it into error, i.e. baptismal regeneration.
Men are named by God to His family from before the world began. We are made aware of that fact, however, when the Holy Spirit regenerates our hearts, usually as young people or adults or even on our last day on earth.
But that regeneration is a miracle, and God can perform it at any point in a person's life. "With God all things are possible."
Because ultimately it is not faith that saves us, but God's grace through faith. And that grace is specific, targeted and effectual for all that God ordains.
And yet no children are mentioned.
The assurance of salvation is made to believers. Was Esau saved? Was Cain saved? They both were part of a family that had members who were deemed righteous because of their faith.
Baptism is always after Belief.
Acts 8:36 Now as they went down the road, they came to some water. And the eunuch said, "See here is water. What hinders me from being baptized?
Acts 8:37 Then Philip said, "If you believe with all your heart, you may."
I don't believe your skepticism over this fact precludes your children from being members of His covenant family. They will come to faith because they've been raised by a God-glorifying father and mother who may very well attribute their children's faith to their own good and righteous choice made after they are baptized.
Why didn't Esau come to Faith? Why didn't Cain come to Faith?
Rome baptizes children.
All the state churches did and they are wrong.
Which brings the discussion back to what roamer_1 and I started with. The Reformation Churches may be a transitional group with the Christian Churches that follow returning to the Pre-Nicea model, or more appropriately the Apostolic Era model. The Reformation Churches are declining just as the Roman Church is and the Christian Churches that are growing are Non-Denominational and Pentecostal. I don't believe infant baptism is a common practice in these Christian Churches.