Another reason could be related to his personal growth. He might be trying to master the power of the eye in private. Naruto has always been one to face challenges alone at first. By hiding his eye, he can focus on understanding and using its power without distractions or the pressure of others' expectations.
Well, Thor can lose his eye in various ways depending on the specific comic arc. Sometimes it's due to a powerful magic or a surprise attack from a formidable foe.