Couple thoughts:
1. If the existing isn't the same as the wheel/tire combo you have, would you, in the event of a flat, use that spare? If not, remove it.
2. If you were to add the same wheel/tire combo as a spare, do you know what the weight of that wheel/tire combo is? The reason I ask, if you're in the middle of nowhere, do you have the physical ability to hoist that spare off the tailgate, lift it onto the hub and hoist the flat into the rear of your JK?
Case in point, I was working on my tailgate/spare mount a couple years ago and had to remove the spare (Toyo MT with Moto Metal 962 wheels). The wheel/tire combination was around 108lbs. I told my wife if she ever flatted my JK to call AAA because there's no way she is gonna manhandle the existing tire off the hub or be able to remove the spare.
I'm not the guy saying, "girls can't do it," I'm just trying to be realistic and focus on practicality. I'd say the same thing if you were a guy asking the question. I also happen to think it's a matter of personal preference if you want to carry the extra weight of a spare or you prefer a different aesthetic for your JK.