Inconvenience was far from the biggest flaw the old covenant had. It was external, rather than written on our hearts, and never really sunk through. It was unable to save and was never meant to save because it was about actions, not the heart.
The American government exists to protect rights.
We hold these truths to be self-evident:
That all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among these are Life, Liberty, and the pursuit of Happiness.
That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed.
Protecting rights will often overlap with morality, but not always. Going to a monster truck rally instead of visiting your dying grandmother in the hospital is morally wrong. Killing someone is morally wrong. Eating excessive amounts of unhealthy food is morally wrong (the body is a temple and all that). Should the government punish all of those? (note that I didn't ask if they should punish them equally)
And I'm pretty sure the early church was still pretty immoral even under the new covenant. You're always going to have to deal with human nature.
True, but notice that in I Corinthians 5, where Paul writes to a church where "it is actually reported that there is sexual immorality ... and of a kind that is not tolerated even among pagans, for a man has his father's wife," he never makes any appeals for the government to do anything. He rebukes their immorality and demands that they excommunicate the guy sleeping with his stepmom, but never calls for a civil law against it.
In fact, in chapter 6, he rebukes Christians who rely on the courts and the government to settle disputes rather than dealing with them within the church.
There is still a need for moral guidelines, and for consequences for breaking them, but not from the government. Under the Constitution, the government's job is to protect our rights and give us security, and nothing in the Bible calls on it to go farther. Rather, it calls on us to do it ourselves.