Why does Goldstein's derivation of Keplerian orbits from Newton's law of gravity not yield the orbit consisting of symmetric oscillation along a line passing thru the center of the Sun?
The fact that an orbiting body cannot penetrate the sun is irrelevant here. By Newton's theory, the oscillating body should accelerate to infinite velocity as it approaches the center, but then immediately decelerate symmetrically on the other side. The period will obviously be finite. This orbit is a maximally eccentric ellipse, and its foci are at the ends, but the sun is at the center!