Personally, I allow both specializations from the Core Rule Book and Run & Gun. Granted, that's not legal for Missions, but it doesn't break anything, in my experience.
Now, if you were to use only the ones from Run & Gun, I would still allow the one for Cyber Implant Weapons, simply because fighting with those weapons is different than using Martial Arts.
As for why you would use Unarmed Combat, if you are holding a blade, you have your hand closed and a different stance, as well as using different muscles. If you have a blade implanted in you, you aren't holding anything and you are striking as if you were just using your hands. It's basically like boxing (not the Martial Art) or just punching, but instead of hitting with your fist, you have blades sticking out. Considering that the specialization was available in 4th, I wouldn't rule it out, but I think it fits under Unarmed Combat better.