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.
Good answer
Amendment I
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.
I agree, there is no language specifying a "separation of Church and State" in the Constitution. It only prohibits any law governing an official religion, or limiting freedom of religion. It appears that we have failed in respecting the freedom of religion, at least in respect to any freedom to govern or legislate in accordance with the politician's concept of God's will. The ACLU will strike it down if it has any reverence to God.