Still have to account for the original timeline where NO time travel occured, though.
No you don't. The original time line has the time travel in it! That's the whole point of the closed loop... there wasn't a starting point, never a timeline where the event didn't happen. Everything is completely consistent. You might as well be saying that you need to account for the timeline where WW2 didn't happen... it doesn't make sense. It always happened.
Going only by the first movie, it was a closed time loop. Connor created himself and that's always what happened and what always will happen. It is the original and only series of events.