I believe Thomas jefferson was in error because the wall keeps government out of the churches but does not keep churches out of the government.
Keeping churches out of government stops any one church from using the power of government to disadvantage other churches. That's the whole point: freedom
of religion is aided by freedom
from religion... at least in terms of the influence of religious institutions and organizations on government.
The American Revolution happened in the wake of (and in many senses was inspired by) the English Civil War. The American Founding Fathers were very familiar with a history that showed that when Catholics gained power, Protestants were oppressed, and when Protestants were in power, Catholics were oppressed. They created a system where no church was in power, so that no church would be oppressed.
Also, in 1776, the states had a mix of approaches to religion: some were officially Anglican, some were officially Congregationalist, some had no official religion (but in the case of Maryland, was predominantly Catholic and very sensitive to the potential for intolerance by Protestants). The Federal level, i.e. the level to which the First Amendment originally applied, the intent (IMO) was to create a system of government that assured all the states that no state's religion would have undue influence over the others. To do this, they engineered a system where religion does not have influence over the federal government, other than the influence of individual voters' consciences. With the Fourteenth Amendment, this system was extended to the states.
TL/DR: you're completely wrong.