So of course both of these are slight simplifications, but what is the connection between the two? If the earth is basically a circle, is an ellipse just a parabola stretched around a circle? Is a parabola just an approximation of a tiny part of an ellipse? How high do you have to be before you change your calculations of a trajectory?
The Math ain’t mathing.
Well, exactly, in the context of OP’s question we’re doing “spherical cows in a vacuum”. In reality, no orbit is truly elliptical, either. Then again, if you use a practical measure for a practical situation, they’re close enough to elliptical for most space travel purposes, and definitely stable far away from an atmosphere.
General relativity can also cause noticeable departures even in our solar system, like the precession of Mercury’s orbit. In extreme situations it can get really different - gravitational waves remove energy, and around a black hole there’s a region where escape is possible but not any orbit.