God never got “put in schools” by official readings, so their absence couldn’t take Him out.
God is not like a fluid or gas that we can pump.
I agree with your general statement(s)
By taking God out of anything I was inferring that we didn’t seem to have these ‘problems’ while prayer was allowed in schools, mangers or menorahs in the town square, calling Christmas Christmas - etc etc
I have often said that if they were to remove ‘under God’ from the Pledge of Allegiance, I could live with it because I learned it the ‘other’ way’.