The problem, generally, is the U.S. was then and now a country of many different faiths, and in many cases the law reflecting the priorities and beliefs of one required the law to conflict with the priorities or beliefs of another. You say Jefferson couldn't be bothered addressing the 'spirit of faith' but then how could he? Whose faith? Which faith? And the faiths as he knew them don't reflect the faiths present in the U.S. today.
You appear to want to condemn "low church" people and their beliefs, but why should your preferences matter? And who decides whether the "low" or "high" church has a more favorable set of sacraments versus some standard that you can't define?
The principle in this country is simple enough. All faiths are welcome and no faith should have the 'right' or authority to impose its values on those of other faiths. It's worked wonderfully on balance. On the same short section of a road near me, you have Catholics, Baptists, Methodists, Jews, Church of Christ, Seventh Day Adventists, Presbyterian, Greek Orthodox, and Unitarians, along with at least two other 'independent' churches. Just off that road is the Episcopalian Church. Every weekend all those churches fill up with people peacefully exercising their religions. That's what Jefferson sought and he succeeded.
Sounds like a great argument for separation of church and state!