if it is bad enough that it still draws the eye after a good sanding and painting, then you have 2 choices:

1. replace the door. this can be very expensive.

2. put a kick plate on it. this is the more popular fix but some door types it is hard to find one that looks right.

First, solve whatever is causing the splintering. If the door is dragging on the threshold then you probably need to trim the door. The two should never touch.

Since the door is painted, you can fill in the missing wood with bondo or a similar filler that can be sanded to shape and repainted.

If the door isn't rubbing too badly, I'd sand a slight 45 degree angle off the bottom corner of the door (google "chamfer" if that description doesn't make sense). Then paint, and be sure to seal the bottom of the door as well.

If your threshold allows it, you could put a weatherstripping door bottom that "saddles" the door if the flanges of the door bottom will cover the damaged area. You will have to cut the door down to the appropriate height to allow for adjustment of the door bottom to the threshold for a good seal.

Thanks for the suggestions. I think I'm going to try filling in the missing portions, sanding and repainting it. It is otherwise a great door, and I would not want to replace it - I don't think I could find another like it.