A carpenter, who usually trims as well, would not have left a gap that must be filled later. That is how is usually works. Your problem is not obvious to me. If your problem is that black strip under the door, you can paint it.
Sand with steel wool. Black paint can stay, just light sanding. Clean with alcohol and let dry for a few hours. Then paint. Spray paint stays in storage longer and has more uses later. Spray some into a container while the top is more or less covered. When you have about an ounce, paint with small disposable brush. Foam brush may melt. 2-3 coats.
That is a hard plastic threshold mounted to the extruded alum sill below. Yours probably has a 3/8" or so thick wood buck under the alum sill, which is where the threshold is stapled to. So if you were to add a moulding to cover the plastic, nail it into the thin wood rather than the plastic.
What I do is make a transition out of oak stock to not notice the height of the sill so much. Nailed to the floor, not the door. This house the prefab doors one had a nicer woodgrain threshold, one black plastic like yours.
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