Honestly, I'm not a biology or chemistry major. It seems that you are more knowledgeable about the technicalities of this stuff. But to out right say that it is impossible to repair damage isn't right.
Again, I'm not a biology major, but you said that you need to know the original DNA code in order to repair damage? An entire genome is really not that much information. And it wouldn't have to be stored directly on the nanomachine. For example, the information needed could be transferred wirelessly. Or maybe the info could be stored directly on the nanobot, I'm not sure exactly and I don't know how you can be so sure, since nanotechnology is only in it's infancy.
Another possibility is to replace our biological cells with synthetic cells. They already have artificial red blood cells that function in animals and are thousands of times more efficient.