I think it is beautiful the way it is, maybe put some light colored wall paper on the back of the shelves to lighten it up.

Please don't paint it. It's charming and perfect just as it is, but painted it becomes just another piece of DIY-painted furniture that is so prevalent as to look mass produced these days. (Trying to dissuade you! 😉)

My thoughts are to keep it original and brighten the inside. We do have the missing piece as well.

Brightening the interior is a great idea. You could easily experiment with wallpapers or even fabric on the inside to bring the eyes inward.

I would leave the whole cabinet. It is a family heirloom furniture. If you want to lighten the inside, then use wallpaper, since it can always be removed and the glue cleaned off.

Once you put colorful items of a substantial size on the shelves, you will have color to attract the eye. Even a collection of white items would draw attention. Nothing black or grey will do. There is no hurry, rather, because this is a cherished memory piece, it should showcase items of family value.

I would stack lovely white linen in there. It's beautiful.