True those exact words are not found in the Constitution.  However starting in the 20th Century the US courts have interpreted the non-Establishment clause in the First Amendment as mandating a church-and-state separation.  And don't forget the Constitution thus mandates the states obey the laws (and by inference, decisions of SOTUS - cf.,  Madison v. Mulberry) of the national government.  Moreover, most if not all state constitutions have non-establishment clauses.  Where did you get that notion about Jefferson who was a Deist and not a great lover of  organized religion?  Ever see his comments about Jesus Christ?