Doug Wilson writes:
“When it comes to child-rearing, between the Old and New Testaments there is total and complete continuity on the subject of godly parenting. There is no discontinuity. It needs to be emphasized again that there is continuity in the promises of God with regard to parenting. Not surprisingly, this has ramifications for the subject of infant baptism” (To a Thousand Generations, p. 10).
I am currently reading this book. Lots of good stuff in there, even for a Baptist. BUT…
Raising godly kids is a Covenant responsibility (or ‘Ethic’). But I don’t think there is a direct link between the Covenant sign and godly parenting. Unless, under the New Covenant, we are now also required to parent our little girls. The sign is for the mediators of the Covenant. Males mediate a succession of flesh. Warrior Bridegroom. Now Spirit-filled males and females mediate a succession of Spirit. Warrior Bride. 
This doesn’t mean there are no promises for the kids. The sign highlights the promises, not the kids. Among other things, a Covenant is a shelter.
The Old Covenant promised salvation through a manchild. It came. The New Covenant promises resurrection for the converted, which is what baptism pictures. Better promises.
The Old Covenant was for the immature: structure and discipline for children, for the born. In some sense, our kids are still under the Old Covenant, or what it pictured, until they are born again. Children of flesh are symbols of children of God. Forming, then filling; natural, then spiritual.
Reviving an emphasis on godly parenting is crucial, and highlighting the efficacy of baptism is too. But conflating parenting (a Covenant responsibility) with the Covenant oath only causes confusion. Ask any ancient Jewess.
Pic: Christina Ramos Art
 New Covenant baptism has more in common with the Nazirite vow (Numbers 6) than it does with circumcision. “When either a man or woman…” (Numbers 6:2). Baptism is for warriors:
“But when they believed Philip as he preached the things concerning the kingdom of God and the name of Jesus Christ, both men and women were baptized.” (Acts 8:12 ); “I persecuted this Way to the death, binding and delivering into prisons both men and women…” (Acts 22:4)