In this age, marriage and family rest at the center of God's plan for social order in the world.
When God created humanity, He created a married couple. Then God gave them an explicit command to fill the earth with children. Genesis 1:28-29, 2:14-24. God re-affirmed that commandment with Noah and his children (Genesis 9:1). God said that singleness was not good for Adam. Though singleness can be God's will for some Christians, it isn't the norm. The norm is marriage, and, as God enables, children. God assigned the highest penalty possible, death, on every form of sexual behavior that deviated in the slightest from the standard of marital monogamy (Leviticus 18).
God always blessed the children of those believers with whom He entered into covenant. God blessed all the children of Noah (Gen. 9:8-17). God promised blessings galore to the children of Abraham (Gen. 12:1-6, 15:1-5, 18-21). God blessed the Israelites through the mediation of Moses (Ex. 3:6-10). God blessed David's male off-spring, through His covenant with David (2 Sam. 7:11-16).
How did Jesus Christ regard children? He held them up as examples of saving humility and faith (Mt. 18:1-4). He promised to severely avenge them, should anyone lead them into sin (Mark 9:42). He demanded that we treat them well (Mt. 18:10-11). He said He came here to save them, the way a shepherd goes to great lengths to save one little lost member of the flock (Mt. 18:12-14). In fact, God's kingdom already belongs to little children (Mt. 19:13-15, Mk. 10:13-16).
How do the apostolic epistles speak of children? They speak little, but this should be understood as building upon the pages and pages of prior revelation that God had already given the prophets, and spoken through Christ. The apostle Paul says that God considers the children of a Christian holy (1 Corinthians 7:12-14). Just as the children of the Jews were "clean" (that is, set apart unto the Lord, by the Lord), the children of Christians are "inside the camp" in the same way.
This does not mean they are saved, or that they can vote or take communion. But it means God regards them as included outwardly as part of the local church, unless they cast off Christ.
The child of a believer is obligated to learn and keep all that Christ has commanded us, whether it is a re-affirmed commandment from the Old Testament (Ephesians 6:1-3, Colossians 3:20), or distinct New Testament truth. We Christians are raising our children in the nurture and admonition of the Lord Jesus Christ (Eph. 6:4), which is how we apply the Great Commission to our own children.
The apostles thus treated the children of a Christian congregation as members of that congregation. Just as a husband (Eph 5:25), a wife (Eph. 5:22), or a servant (Eph. 6:5) were spoken to as part of that congregation, so was the child (Eph. 6:1). A Christian's child is not some unclean interloper, straggling in from the wild outside world. Paul greeted entire households in the Lord, not just the adults (2 Timothy 4:19). John greeted the elect lady and her children (2nd John 1), even though only some of them were walking in the truth (4). I do not see any internal evidence in 2nd John to take these terms as symbolic.
The Bible's clear message on marriage and family is a reason why we should not overly segregate the age-groups in church. I am not one of those who condemn age-grouping for educational reasons, as if doing so is a carnal compromise with Darwinism, Deweyism, and secularism. It is wisdom, not compromise, to recognize the differences in how the various ages learn. (This principle applies even more so to special-needs children, depending on the severity of the disability).
However, I also know I have always attended small-ish churches (200 in attendance or fewer). So there was never any danger, in any of my past churches, of the children having inadequate inter-generational interactions. I have never had to deal with the programmatic blight of big churches sending every age-group shooting off in completely separate directions at every church function. So I recognize from the outside looking in, how this could become an important issue in certain kinds of highly programmed churches.
I am also not one to condemn youth ministry. We slander many fine youth workers, whether paid or volunteer, when we paint youth work with a broad brush. Yes, some youth ministries are run like a religious youth compound, sealed off from healthy adult relationships. But children also need something Christian for them, and not be forced to sit around waiting while we adults suck down coffee and boringly chatter with each other. A rounded youth ministry should always be a family ministry. When we minister to a child or a teen, we are still under the parental authority. We are also seeking to make an impact for Christ on the whole family, if God allows.