The "original Egyptians" were not Christians. That's for sure. And their culture was nowhere near the same as modern day Copts.
And IIRC the etymology of the word "Copt" is Greek. Why would the natives name themselves after a foreign name? If there is a continuation of legacy then why not keep the same name they always used from time immemorial? It is also my understanding that most Copts are descended from people who came in from the Middle east during Roman rule i.e. middle eastern Christians. They mixed with locals to some degree, but the the movement is outward, to inward.
IMO the people closet to the Ancient Egyptians are those Egyptians found in the rural area or Upper Egypt, especially the Fellah's. These people: