While the constitution did not specifically name the Christian religion directly, I don't think that were directly speaking of any religion, but the different Christian religions which had left in England when the original colonists left. There is a difference when your religion preaches destruction to other religions, because others do not believe as you do (Islam) would have in no way been endorsed, therefore the separation of church and state means that government cannot dictate which form of religion (Christian) the citizens are to adhere.
That is where the difference is in my belief and those that don't think the Constitution was referring to Christianity. Otherwise Satanists would be allowed to practice their religion which calls for sacrificing humans including babies, so you think all religious activities of all religions should be tolerated?
This is one of the reasons we need a wall, as we as a country have sovereignty over our borders whether we trade with the neighbors or not, and with a neighbor like Mexico, who has allowed others from other countries to access that open border. The US should withhold all funding, just as they have allowed the corruption in their government to be overrun with drug cartel money to allow the drug trade to flourish, shame on US for having succumbed to the drug addiction problem, but that is not a reason to allow it to continue !