I am an ophthalmologist also. Maybe I can help you.
You are correct in that lack of sleep is a very rare and temporary cause of dark under eyes. Most causes are from malnutrition, allergies, and hyperpigmentation. However, the most common cause is from leaking of small blood vessels just under the skin. In essence, it is like a bruise (a persistent one).
Treatment depends on the cause (i.e. malnutrition requires vitamin supplements, allergies require treatment, lack of sleed requires sleep, etc.). As far as the hyperpigmentation, there are a variety of lightening creams and makeup. Chemical peels and dermabrasion have been tried as well. For the blood vessels, lasers and filler injections may help.
Hope this helps and thank you.