tt40 wrote:
In everything I've read about planets and elliptical orbits, I can't
ever recall any author (Feynman, Newton, 'Ask an Astronomer' etc.),
explaining exactly 'why' the orbit is elliptical. Oh sure there's been
lots of mathematics to explain the orbit and how it works, but most of
the explanations don't provide a definitive statement as to why it IS
elliptical.


The answer lies in mathematics of Kepler, and especially Newton.

Johannes Kepler -- more than 900 pages of calculation in about four
years attempted to measure the orbit of Mars

o from a moving platform
o who's orbit was not centered on the Sun
o rotating on its axis
o of a planet varying in its orbital speed around the Sun.

Kepler's calculations were immensely difficult... but he eventually realized
that an ellipse with an eccentricity of about nine percent agreed with
Tycho Brahe's observational data. Kepler found that the orbits of planets
in our solar system follow ellipses, sweeping out equal areas in equal
times (Newton would later show this as the conservation of angular
momentum as you point out).

Newton discovered (and showed mathematically) that objects in free fall
(such as planets influenced by a central force like the Sun's gravity)
follow the paths of conic sections.

Ref: http://learning.physics.iastate.edu/DemoRoom/MU.htm#22

The combination of Newton's law of gravity and F=ma . The task of
deducing all three of Kepler's laws from Newton's universal law of
gravitation is known as the Kepler problem. Its solution is one of the
crowning achievements of Western thought.

A model for Gravitation was essential.
http://scienceworld.wolfram.com/physics/Gravity.html