After a long pause, I decided to go to the original ID sources and get it working on the 16MHz Atari Falcon030, with minimal changes to the C code. This is the result!
This is using ID's original software raster, as compiled C (using BrownELF GCC6). There is no assembly code at all, except to convert the framebuffer. Obviously this is going to be super slow.
I'm currently trying to refashion my own rasterizer (seen in the other videos) to be able to talk to the real game code, so the components can be swapped over. This probably won't be too easy since its not derived from ID's engine code but will see how it goes...