Just like the title says... any way to make this look less bad?
Paint it, or pull off the hardwood edges, veneer over it with a new piece of veneer of the same species, and then apply new edges.
Depending on if you have multiple other matching finished end conditions, you could create a new trim detail that covers over the affected areas.
It looks like he face screwed, and then filled the holes with epoxy filler or possibly plugs of some sort, and then did an abysmal job of sanded them flush.
I work for a commercial millwork company, and if one of our installers did something like this, after firing them, we would replace that finished end panel completely.
One other possible solution is to apply a decorative trim pattern that covers over the screw/blemish locations. I would begin by sanding the entire surface to provide a surface to accept glue and new finish. Then glue and clamp new trim over affected areas using trim in a profile and design that compliments the project's style. You might also use fine gauge finish nails to reinforce the attachment of the trim if there is difficulty clamping it. Clean up any glue squeeze out with a wet cloth immediately. Let it dry, lightly sand and apply new finish.
Oh yeah, finally, I would offer to repair the contractor's furniture using the same techniques used on yours:)
Already proposed re-veneering is a good idea but not easy (removing edges) and veneer is probably difficult to find. My ideas involve some cheating:
Get some artist pencils with suitable colors and try to create missing grain back. Then spray shellac from a can. I am not sure if during finishing area should be horizontal, but that wont hurt for sure. Those darker dowels are probably more difficult to fix than brighter areas.
Make a good photo of veneer from other areas, print and glue into whole area inside edging. Maybe some moisture resistant paper + wood glue or some self adhesive paper?
Good luck, it is not an easy fix.
Just a thought on the cheating side: Might those infills be colored beeswax? If so, might you be able to melt/syphon some of that muck with an iron and a cotton cloth? At least enough to backfill with a more suitable base color. Then you could apply @omark's fix with pencils. We use oils, ourselves, to add grain or knots to minor blemishes. But the operative word is "minor", like a blemish in a library full of panels. I cannot imagine lying on my side for hours fixing your panel - but everyone is different, and that could be an appealing solution. I mean that without a trace of sarcasm - I am always amazed at how things I detest can be found satisfying or beautiful to others.
@Omark's other idea definitely has merit, if you can't find that veneer.
New veneer would definitely be the best solution if possible. You might also consider a laminate you can apply only to the sides on both ends, balancing the piece and making it look like the original design was to do that. Another alternative would be to lacquer over the side panels, though if you've never applied lacquer before be sure to practice on some scrap - it can be hard to get a good finish and you'll want to either sand/fill or remove enough of the patch your contractor applied to restore some amount of grain (any open grain on the wood will show up on the lacquer coats). If it were me, I'd remove the end panels and remake them (sorry!).