Is it ok to let laryngitis "run its course"?

Yes: Laryngitis is a viral illness and there really is no direct treatment to shorten it's course. Antibiotics do not work.

Laryngitis: Laryngitis is an inflammation of your voice box (larynx). It is usually due to an infection, most often a virus. You may become hoarse but symptoms usually go within a week or so. See a doctor if symptoms persist. http://patient.info/health/laryngitis-leaflet

Laryngitis : Laryngitis is usually caused by virus or by voice abuse ( shouting , screaming ) . Usual treatment is voice rest , steam inhalation. Antibiotics don't work for it and laryngitis takes its own time to get better.