However if I want to draw my sphere using circular wireframes, that projection-cone is no longer a circular cone. So it's not your traditional conic section anymore.
The perspective projection of a 3d circular disk is also an ellipse, as long as the disk is entirely in front of the camera. (If the disk crosses the camera plane, it projects into a hyperbola).
It is only slightly more complicated to compute the ellipse parameters of a projected disk than to compute the ellipse parameters of a projected sphere. And as you've pointed out, there can be greater utility, efficiency, and accuracy in using curve primitives for rendering when curve primitives are available, especially when using a vector based renderer like SVG rather than raster based renderer.
There are several methods for computing the ellipse parameters of a perspective-projected circle, using geometric constructions, or using analytic / algebraic methods. I've found the algebraic methods easier to implement.
Perhaps the simplest algebraic method is to put your disk into matrix form , run it through the perspective transform , and then extract the center and axes from the result . A simple alternative to the first two steps is to compute the projected ellipse equation by taking 5 sample points from your disk, projecting them to the image plane, and converting the 5 points directly into the ellipse equation coefficients , then proceed with extracting the center and axes from the coefficients .
There is a nice complete code example of computing the ellipse using a transform on ShaderToy  which has a blog post attached describing the math and an interesting application of it, undistorting text. 
One constructive geometry method is to perspective-project a square that encloses your 3d disk, and find the ellipse inscribed by the projected quad . It uses known geometric relationships between the quad and the ellipse (such as the tangents of the contact points) to find the ellipse parameters. The geometric relationships of parts of the ellipse have been explored in depth 
And just in case you want to go the simpler route of drawing the sphere's silhouette ellipse (rather than wireframe disks), Inigo Quilez wrote a nice article deriving the math  and he provides a complete implementation .
 https://archive.org/details/constructivegeom00eagluoft/page/96/mode/2up (pages 96-150)