New Englanders, like John Adams, felt that State should care about religion and religious education, requiring towns in MA to pay salaries to preachers. He, and others, felt that society needs morality. So, it was not about having trees or menorahs at town halls, but about having middos.
The controversy was whether they should all be from the most popular sect, or could local hasidim redirect payments to their own shtibles. Interesting, this is about the times of Vilna Gaon and Chasidism controversy. Adams was definitely a Litvak….
I think we all agree that we benefit from US gov not running a church, but does it mean that we support excluding G-d from public sphere? “Sorry, G-d, we find it easier to do Your Will if we keep it quiet somewhat”? This seems like throwing the baby with the water.