And yes, you could remove "In God We Trust" from the money - I can't say as having it on there particularly adds to its value.
You are on the "clean it all out" side. At least you're intellectually honest and don't deny the presence of the elephant in the room.
However, acknowledgment that it should be cleaned out is also acknowledgment that there IS "God talk" in official places/documents/instrumentalities of the U.S. Government.