I did see that the edges aren't coplanar.

The way to start out with all your edges coplanar is draw a large rectangular face on which to draw the profile. Then delete the geometry that is not the profile.

There's 1001 bit shadows and sky ratio:
http://www.1001shadows.com/en/products.html

Maybe there are other tools, too, I have never needed them. It's a commercial plugin.

Jean- thank you so much for that answer!

Between that and using "intersect with selection" rather than "intersect with model", I'm now getting exactly the results I expect!