The DXF you generated works fine. I will try your steps.
This part was drawn as a 3D object. Started with a cylinder and built from there. I have also tried starting with 2D shapes and extruding them, either way I end up with faceted sides. What I do notice is, when I draw 2D, the outline is smooth, circle, arcs, whatever. As soon as I do the extrude, the outline becomes faceted. This may just be from lack of knowledge on my part.
Your file demonstrates it can be done, I will try both ways you mention and see how they work, maybe one will stand out as easier.
EDIT: Your mention of explode got me to go and do that on my DXF that won't load. After doing an explode, the DXF works fine and produces two continuous curves for the cutter software, looks just like the DXF you produced.