Misunderstandings are the normality, especially if one strives to be concise in dealing with such arguments.
My first reaction was against your statement, according to which centrifugal forces appear only in circular orbits (didn’t you say that?). Yesterday, I prepared one page in WORD to recall the basics of kinematics about plane motions, to show, by use of the appropriate equations, that a point in any non-linear plane motion undergoes centripetal accelerations; therefore, this is true also of any point moving along orbital paths, with a particular characteristic if the motion obeys Kepler’s laws. This characteristic consists of the existence of a centripetal acceleration that is constantly directed toward the center of motion (i.e., the Sun for planets), this center being also the origin of the fixed reference frame. Unfortunately, I wasn’t able to transfer my memo and relevant formulas in this reply-window of the Forum, but any good manual of analytical mechanics can work much better.
My second point (referring to Newton’s second law), was the obvious remark that if the moving point has a mass and undergoes an acceleration, the material point is subjected to a force. Do you disagree? If so, I cannot comment further.
Surprisingly, you now state that the combination of centripetal and centrifugal forces should have nil effect. Instead, just to begin, I think you should know about the existence of “tension” and “pressure” effects. By the way, in the case of planetary orbital motions it regards “tension” on the planets, which, according to scientists, should also be viewed as the best explanation for tides, whose “strange” characteristic is that tide effects (on the Earth it's a well-known experience) appear symmetrically and simultaneously at opposite geographical areas of the Globe.
The main point, however, is that gravitational attraction between the Sun and the planet (let’s limit to the two-body attraction, for the sake of Newtonian simplicity) establishes a mutual dynamical constraint, quite analogous to the force transmitted by a band to the stone in an ancient sling weapon. In the sling’s case, the stone orbits the hand that holds the band, thanks to the dynamical equilibrium established by two opposite forces, the one transmitted by the band (which is a “constraint” analogous to the effect of the mutual attraction between two celestial bodies) and the centrifugal force the stone undergoes because of its constrained “revolution”; (in this connection, among other things, note that the stone’s orbit cannot be considered as “circular”, since also the holding hand rotates. In such devices, as you know, the band may lacerate if the stone's revolution speed exceeds a certain limit: is it not the tension due a centrifugal force?). Strange indeed that I’ve to ask this question.
Therefore, once ascertained the existence of a centripetal force acting upon a MASS IN MOTION, an opposite centrifugal force must also affect the same mass: it’s an every day experience for every one in the world, especially when the vehicle in which we travel is compelled to swerve. Is it not so? Your interpretation of Newton’s third law seems peculiar.
Yet; the initial topic was about the derivation of Newton’s laws from Kepler’s, and/or vice-versa.
Newton was able to prove that the planetary orbits, because of the Kepleran laws, are characterized by a centripetal acceleration that is constantly directed as the straight line connecting the planet to the Sun. This was the key point proved by Newton to obtain his gravitational law.
(In my view, your mention of polar/equinox precession, or Coriolis forces, or rotating reference frames etc. has nothing to do with the point in question, and does only contribute to create confusion).
By a very quick summary, and sticking to kinematics only: in any plane motion, as possibly described by a fixed polar reference frame, the point in motion undergoes in general an acceleration A, which can be thought of as resulting from two component accelerations AR and AT, dubbed “radial acceleration” and “transverse acceleration”, respectively.
Radial acceleration AR points constantly to the center of the motion (i.e., toward the origin of the reference frame along the vector radius), whereas the direction of transverse acceleration AT is orthogonal to the former.
It can easily be proved that transverse acceleration AT is nil (i.e, AT = 0), if the point moves according to any orbit that obeys Kepler’s laws. Thus, the only acting acceleration in Kepleran kinematical orbits is the so-called “radial” or “central acceleration” AR . From which all subsequent considerations.
As to precession, I didn’t misunderstand you. I have deliberately mentioned the perihelion precession to mean only that Newton’s gravitational law is an excellent though approximate model, which is not sufficient for an adequate explanation of a number of astronomical “anomalies”. The reason is just that Newton’s law is a direct logical implication of Brahe’s observations and Kepler’s laws.
