Have you tried Inkscape ? It is Open source, hence free, like Ben said.
I use Illustrator, because I have been since version 2 or 3 and i am used to working in both Illustrator & Rhino, to do 2D lines for concepts. I also use Adobe Line & Draw on the iPad which makes it even better to work out ideas while on the road and know I can access it from home or office. Not sure if these are free.
The converted added is because Rhino exports to an older version of Illustrator, not sure which one.
So maybe you should be modelling with Blender, it’s free and OpenSource
It sounds like it is the way Affinity Designer is importing the AI files. I have never had issues with layers in Illustrator from Rhino.
All the best if finding a solution.