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I disagree with your third bulleted point.
The scientific method must necessarily be discarded when practicing astrology. Although one could arguably make a case for a "logical method" based on a set of accepted axioms, one should not call it "scientific". It is more correctly characterized as a pseudo-scientific method.
The scientific method is hypothesis followed by experiment (or observation). Should the outcome of an experiment be contrary to the outcome predicted by the hypothesis, then either the experiment is flawed or the hypothesis itself is incorrect. In order to apply the scientific method, one must have a testable hypothesis. Thus both religion and philosophy are not amenable to the scientific approach - there are no experiments to be devised by which one can truly test their predictions.
Astrology may be internally coherent, but it makes no predictions with measurable and verifiable results. It may drape itself in the trappings of science in an effort to increase its plausibility but that's makes it nothing more than another cargo cult. I'll grant you that astrology may even be true and that one is certainly entitled to believe what one pleases - but it's not science and the use of pseudo-scientific methods serves only to attract the impressionable, convince the gullible and infuriate a great many others.
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"""Just because the fundamentals of astrology have no roots in science (aside from the computations), doesn't mean that scientific methods should be thrown out when practicing it."""
I don't mean "the scientific method", but more methods and techniques devised by science/scientists. For example, use of statistics and probability theory.
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What would one do with those methods? They'd really be of very little use without a hypothesis; serving only as a scientific veneer over what is uncharitably labeled superstition - truly a cargo cult.
If your contention is that the alignment of the heavens contribute to personality then one should be able to devise experiments to test this assertion. But, without a reasonable, postulated mechanism for the influence of heavenly alignment on personality, one should probably approach the enterprise with a large amount of suspicion. I'm not saying that this heavenly influence cannot be; but surely some bright individuals can apply scientific methods to test whether it is or not. Otherwise it's just "faith".
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Well, my point is not that one should use those methods to (try to) prove anything without a hypothesis. I'm merely saying that statistics etc can be useful in astrology.
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I was going to just ignore the astrology posts but I happenned to scroll down from the cat pictures, and it occurred to me that many of the bulletted points on your list (at least the first five) work with s/astrology/hacking/ -- that there's the "media/public" version of the term, and there's the classic, more precise (in the hacking case, "hacker's dictionary" definitions) one....
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Yah, I suspect there are more topics and terms that would fit here...