So rules-as-written, the necrograft arm does not seem to “inherit” anything about the replaced arm. The restrictions on the vestigial arm do not apply to the necrograft arm. And everything the necrograft arm explicitly gives you, the increased lifting capacity, the slam attack, and the “clumsy” penalties, you get those (you take the penalties even if you use your two remaining perfectly-good arms and hands and keep the third, necrograft, arm out of things).
The difficulty comes in determining what functionality of an arm you get that are not explicitly mentioned. The rules text never says anything about the arm behaving like a normal arm. There isn’t even a general statement that necrografts act just like the organ they replace except as mentioned. But this is certainly implied—if it weren’t so, the necrograft arm couldn’t really be used for anything but lifting or slamming, making the “clumsy” penalties rather redundant. And as the vestigial arms discovery itself shows, Paizo does tend to indicate when an arm cannot be used to swing a weapon—in normal circumstances, I certainly would have every expectation that the necrograft arm could be used to wield weapons.
But it’s not really fair to call that the rules as written: after all, it isn’t written, just implied. Thus, rules as written, we can say that the necrograft arm can lift and slam, and gives you penalties doing certain things, but beyond that the rules as written are simply deficient. Whether or not you can do other things, including swing a weapon, are simply undefined or left in a vague, implicit state. You’ll have to ask your GM.
Who will, most likely, decide to apply all the restrictions of the vestigial arms discovery to the necrograft arm, since it clearly isn’t intended as a workaround to those limitations, and is not priced accordingly.