Basically, from about the 1770s there was lots of arguing among protestant sects about how to defend their turf from Catholics (unaffectionately referred to as papists or, my favorite, The Forces Of Popery).
This was a low boil until the mid 19th century flood of Irish immigrants (and later Italians). Protestants couldn't organize anything at scale against Catholics because of dogmatic differences.
Midcentury a consensus emerged that's more or less what you said. Replacing Christian Dogma (which included a crazy diverse set of mostly internally consistent belief systems widely contradicting each other) with Family Values. Since that's basically what they agreed on anyway, it was the obvious place to build a ruling coalition. Another factor in this was trying to heal the split between pro-slavery sects and anti-slavery sects. Whittle away that dogma so civility can be preserved.
This has had some profoundly weird long term effects. First, American Christians who aren't Catholic or Orthodox don't know shit about their own religion. Second, Catholics won the right in the late 19th century to run their own schools because they faced discrimination. Third, this accelerated Enlightenment secularization modern Christians loathe because they were trying to choke the spread of other Christian sects. Fourth, the creation of "non-denominational" Christian churchs as the forerunners of content and challenge light megachurches.
All this had the same effect as the Roman colonization and conquest of the Zealot Yeshua Bin Yosef into the head of the Roman Imperial cult starting with Paul through the adoption of Christianity as the Roman state religion.