I took a look, and at least part of your troubles here arise from a somewhat hidden feature that allows you to select a planar sketch and use that for defining the wrapped toolpath.
This is handy if you want to do fully wrapped features, or if you want better control over the geometry (as well as proper arc support).
In this case, however, it doesn't detect that the sketch is a 3D sketch and already wrapped around the geometry.
Fortunately, this is user controllable albeit the controlling parameter is currently only exposed in compare and edit.
So one option is to use your 3D sketch with the projected lines, and then after creating the operation go into edit and compare and set "Unwrap Geometry" to Yes:
image.png
Alternatively, create a 2D sketch and use that. Ideally, draw the sketch on a plane tangential to the wrap cylinder.
I hope this helps. In the meantime we will think about ways to make this less hidden.
/Mark