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This conclusion was reached using a potential that depends only on the distance between the two objects. In that case (no external forces), the center of mass of the two-body system shouldn't accelerate. But with one stationary object and another circling around it, this can never be the case. It seems like an approximation in which one of the two objects (the sun) is much more massive than the other, but I don't see that assumption appearing anywhere in the derivation.

The kinetic energy is first expressed in terms of the velocity of the center of mass and the relative velocity of the objects. In polar coordinates the Lagrangian leads to three equations of motion, and filling in the 1/r potential immediately gives elliptical paths.

What's going on?