[Cleonis]

In retrospect: the earliest recognition of a conserved quantity was Kepler's law of areas. Isaac Newton later showed that Kepler's law of areas is a specific instance of a property that obtains for any central force, not just the (inverse square) law of gravity.

About symmetry under change of orientation: for a given (spherically symmetric) source of gravitational interaction the amount of gravitational force is the same in any orientation.

For orbital motion the motion is in a plane, so for the case of orbital motion the relevant symmetry is cilindrical symmetry with respect to the plane of the orbit.

The very first derivation that is presented in Newton's Principia is a derivation that shows that for any central force we have: in equal intervals of time equal amounts of area are swept out.

(The swept out area is proportional to the angular momentum of the orbiting object. That is, the area law anticipated the principle of conservation of angular momentum)

A discussion of Newton's derivation, illustrated with diagrams, is available on my website: http://cleonis.nl/physics/phys256/angular_momentum.php

The thrust of the derivation is that if the force that the motion is subject to is a central force (cilindrical symmetry) then angular momentum is conserved.

So: In retrospect we see that Newton's demonstration of the area law is an instance of symmetry-and-conserved-quantity-relation being used. Symmetry of a force under change of orientation has as corresponding conserved quantity of the resulting (orbiting) motion: conservation of angular momentum.

About conservation laws:

The law of conservation of angular momentum and the law of conservation of momentum are about quantities that are associated with specific spatial characteristics, and the conserved quantity is conserved over time.

I'm actually not sure about the reason(s) for classification of conservation of energy. My own view: we have that kinetic energy is not associated with any form of keeping track of orientation; the velocity vector is squared, and that squaring operation discards directional information. More generally, Energy is not associated with any spatial characteristic. Arguably Energy conservation is categorized as associated with symmetry under time translation because of absence of association with any spatial characteristic.