Why glutathione doesn't whiten my skin?

Because that isn't what glutathione does.

Glutathione can best be remembered for AN IDEA: It is your body's master ANtioxidant, Immune system optimizer, Detoxifier, Energy Restorer, and Antiaging protein.

Taking glutathione pills will not raise your glutathione, and they will not whiten your skin.

Raising your glutathione levels with the building blocks so your cells make it internally will bring you the above benefits, but skin whitening isn't one of them.