At least one thing you say, to me, is not just a simplification, it's wrong. You are clearly working in an inertial frame throughout the post (which I agree with since it is simpler); but in an inertial frame, what you call "tangential velocity" (it would usually be called "centrifugal force"; you incorrectly use the term "centripetal force" at one point) is not a force. A spacecraft in a free-fall orbit (i.e., rocket engine not firing) has only one force acting on it: gravity. That's why it moves in an ellipse and not a straight line.