I need to draw a coil, similar to a slinky, preferably as a brush so that it can easily be made into various configurations (arc, s-shape, and so forth), without distorting the coils. I can almost get that look by drawing multiple ellipses but I can't figure out how they can be coiled together in one long coil.
A brush won't respect perspective.
Do you need perspective? Please show how you want it to look. An example? Your initial drawing?
So I conclude that perspective is relevant.
In that case you can't use a brush, but will need to draw and/or adapt those slings one by one in order to get them into perspective.
This is a very manual, convoluted way to possibly get what you want.
What looks like a circle is actually a Spiral that is spiraling straight down on top of itself.
It sort of works. See attached. I added a Silver Gradient to the Stoke to make it easier to see the overlaps.
Interesting technique, Jeff. It beats drawing every anchor point from scratch. I like it and will give it a whirl.