I had not noticed this thread before, but now have read through the posts and found it interesting. First, I was unable to find any exacting history regarding only flour and water to make the unleavened bread for communion. I thought I might find something in J. R. Graves's writings, who influenced our churches in the South (especially the old Southwest) greatly regarding the topic of the Lord's Supper. However, I have not found anything yet. He strongly advocated unleavened bread of fine wheaten flour, but I did not see he ever specified or excluded other specific ingredients.
I found a recipe for unleavened bread online from Statesboro Primitive Baptist Church in Statesboro, Georgia. It included only flour and water, but no details as to the why. An online story about Macedonia Missionary Baptist Church in Crestview, Florida (a predominantly African-American congregation) getting a historical marker, mentions a lady her demonstrating how her mother used to make communion bread from "a mixture of plain flour and water." My mother made the communion bread for many years for our East Texas Baptist church of only flour and water. All that to say, I don't know how it originated, but it seems to be fairly widespread. My inclination is that it is not related to the Catholic practice/requirement.
We use homemade communion bread at our current church, which I believe has a small amount of oil and salt in addition to the flour and water.
As far as unleavened bread is concerned, a mixture of only flour and water is unleavened, and I would be quite satisfied to not add anything else if the church thought it should be that way. I would not make it (oil and salt) a hill to die on. (I would not budge from unleavened, though.)