This does NOT imply that American leaders cannot honor the ideals of God in the execution of their duties.
The "separation of Church and State" is merely to prohibit the establishment of an American official religion in government. We don't have a Southern Baptist Executive branch or a Methodist legislative branch. We strive to maintain freedom and open minded faith. Those faiths who promote repression and death can expect to have no representation in our government. At least, I hope.
The term is nowhere to be found in the constitution.. What you suggest as an alternative is (not a church advocated religion), but NOT seperation of Church and State" which has become a litmus test.