Your saying the "Wash"is the sloped area at the top of the chimney on the last course of brick?
Even so, there is a slight crack between the flue and the last course of brick. Doesn't it make sense to have that area to be able to expand (hence caulk) and to prevent water from freezing in the crack.
You agree, tar is ugly.
I don't understand the "burlap thing". Is that basically to slow the curing of the cement, buy covering it with wet burlap?
The same as a drip edge keeps water running off the edge of a roof, I suppose a precast "wash" depending on how it's made can throw the water away from the chimney just like a drip edge.