Does the 3rd tail light need replaced?

If you want to pass state inspection if your state has them.

Check your local laws, some states or counties do and some states or counties do not require a third eye brake to be operational if equipped.

Because it's location on hatch, the wiring may be broken inside wiring insulation, sometimes can be repaired if you have minimal electrical skills. For most folks replacement is easiest. And yes if you live in an inspection state Tx WY NY Az Ore,wa they are required to work unless you have income to give crooked courts.

Not in my section of Oregon

Why would you ever want to drive around without the third brake light operable???? Who care if it is "required" in your state. It is an important feature of your car and, like air in the tires, should work for you and may save your life or others riding with you. Get it fixed!