After even more work with a utility knife and a bit of sandpaper, the door is starting to look better:
We might actually be able to keep it natural wood... but there is one problem. The citrus stripper I used to remove the paint from the decorative moulding has left a stain on the wood. I'm not sure if it's citrus stripper, or if the citrus stripper disolved the dye from the red paint that was the bottom layer, but there are red/orange stains on our wood. The stains are somewhat deep; a good sanding doesn't remove them. I've tried mineral spirits and denatured alcohol.