Namaste,
I think many of your arguments are strawmans. Allow me to illustrate:
Chanakya and several others called the Lokayatikas astute logicians. There is absolutely no reason why a Lokayatika would oppose science. On the contrary, of all the darshanas, it is the Lokayata Darshana that would be most supportive of science - as long as it was not about investigating mermaids and after-lifes. When we talk about science, let us keep science as focused on the normal and science as focused on the paranormal separate. Compare this with other religious beliefs, where adherents dislike science and curse scientists for advancing into areas that threaten their pet, age-old beliefs. The earth turned out to be round, Rahu and Ketu were not demons, Indian civilization is not billions of years old. It threatens the very fabric of religion - if all of this has turned out to be false, what is the credibility of everything else that came from the same source? For the faithful, there is a lot to lose and hence more incentive to suppress science. The Lokayatika goes by perception. I don't believe in an afterlife because I have no evidence. Show me evidence and I will believe it. It cannot get any simpler.
Nyaya-Vaiseshiks, Yoga and Samkhya are also not about investigating mermaids and afterlife. They are about investigating the world as it appears to perception, and then from that drawing inferences using the classical effect to cause and cause to effect logic that is peculiar of Hindu philosophy.
None of the claims you are making were made of Rahu, Ketu are demons, Indian civilisations were made by Nyaya-Vaiseshika, Samhya-Yoga. They were made by Puranic Hindus, and Puranic Hinduism is not Hindu philosophy.
You should not put metaphysics in the same category as superstition and fantasy. It is disrespectful to metaphysical philosophers and their actual works(vast majority of philosophers in history have been metaphysicians, including Plato, Aristotle, Spinoza, Bergon etc) Like all philosophy they are using formal logical arguments to make their cases. To argue that to use philosophy to make a case for the soul and reincarnation is not philosophy is nothing more than narrow minded dogma.
Philosophy does not contain any preconditions within it that formal reasoning can only be used for x, but not for y. You can use formal reasoning to establish any conclusion, as long as your argument obeys formal rules of logic. If the conclusion is demonstrated by your premises it is valid.
Likewise, empirical science does not have to be limited to the physical. You can use the scientific method to study anything that is empirical and apparent to us. The scientfic method can be used to study meditation for example, and in fact such studies have been done. Using a combination of quantitative and qualitative research methods we have a huge body of scientific evidence for meditation. Researchers have also also studied rare phenomenon like past life memories using investigate methods in science, OBES, matter-mind interactions. Again there is no precondition in empirical science that you must study x, but not y. Anything that is empirical is open to scientific study.
Inference is not necessarily scentific. The boogeyman was inferred by sounds in darkness. While perception and inference both were considered valid sources of knowledge, it was also recognized by scholars that incorrect knowledge could be perceived or inferred and considerable effort went into filtering knoweldge. Obviously, there was no universal methodology here or we would not have had multiple battling doctrines. Lokayata took the most stable position by keeping inferences to the minimum.
The inference that there is a boogeyman by sound in the darkness is not entailed by sound in the dark, because first of all the boogeyman is a premise which itself is in need of proof(like the sky flower) and there is no relationship of invariable concomitance between sound in the dark and boogeyman. The only conclusion one can infer from actual sound in the dark using causal logic that there is cause in the dark that is causing the sound that is not visible. Various possibilities are possible it is an object falling, there is animal or another human being inside the room etc. In order to resolve this doubt we must examine the quality of the sound and match it the sound qualities that it matches. If the sound is barking then one can infer there is a dog present in the dark, or a human imitating a dog, or a recording of a dog.
Inference was very important to Hindu philosophers(especially Nyaya) so they studied the science of formal reasoning very precisely and what made an argument valid or invalid. The conclusion has to be entailed by the premises, if it is not, the argument is invalid. So very precise attention was paid to proving ones doctrine using very precise arguments and formal reasoning. This is the difference between philosophy and faith. None of the arguments presented in Hindu philosophy are based on faith.
said their positon is Sabda is the only source for the paranormal as that cannnot be determined through any other means.
I am not sure where you are getting this from. They said that things that cannot be known by perception and inference, can be known from testimony. This does not mean that they said soul, reincarnation, karma etc are proven by testimony, in fact on the contrary, they proved these things through inference using observable evidence. What they mean by testimony is things that cannot be known by either means, such as the experience of infinite bliss and love, gods. These things can only be known through supersensuous experience(anubhava) and here the testimony of the Vedas is offered.
As for the proof of mind, soul, karma, reincarnation, other planes of reality, god. For this the Nyaya-Vaiseshika, Samkhya gave logical proofs. The Nyaya-Vaiseshika for example give very extensive arguments to explain why the body is not the mind, why a creator is required to explain the act of creation(Hindu versions of the cosmological argument). The Samkhya gave their proofs of the existent effect in order to prove that matter originally is only potential and all matter is reducible to the root prakriti. The only thing that is not reducible is the observer itself and they explain this by showing a property dualism between the two by citing the properties of both entities and giving logical arguments to prove the existence of the observers and root matter(Hindu version of mind-body dualism)
Nyaya and Vaiseshika are realist philosophers because they believe that the universe is real and has particular entities which are not reducible any further. It consists of the 5 elements, mind, space and time and consciousness.(later was added the category of non-existence) In these 9 primary and unqiue substances abide unique logical qualities which distinguish them from one another, for example the distinct property of mind is cognition, as opposed to the distinct property of earth which is weight and extension. The distinct property of consciousness is pain, pleasure, ignorance, knowledge and desire, as opposed to the property of body which is inert(jada)
Lokyata were not realists because all they admitted to exist was matter and did not believe in any particular entities, everything other than matter was unreal(they admitted only 4 elements, and not time, space, mind, consciousness) They are monists in that all that exists is matter and nothing els. The proof of anything other than matter existing was established through inference, which lokayata did not accept. That is because senses only show matter and nothing else. If you accept just the senses as your means of knowledge then you must conclude all is matter. However, it is impossible to deny that we have means of knowledge other than our senses. They tell us divergent conclusions to what our senses tell us.
The conclusion that consciousness continues to exist after the death of the body is not shown by the senes. The senses appear to show that bodies that were conscious are no longer conscious. Logic shows however that consciousness and body are not reducible to one another. If one ceases to exist the others non-existence is not entailed. This is because body continues to exist after death even without consciousness, thus showing that one can exist without the other, hence there is no relationship of pervasion. If the body is the substance that leads to consciousness than the body must contain consciousness within it, but this is not true for the body can exist without consciousness. It is also inferred that perception only ever takes place if three conditions are present: objects, instruments of objects and observer. The body and senses maybe working, but if the mind is not present, no perception ever takes place. If it only when the mind is present that perception becomes determinate. So what is this entity called mind which can be present or not present, and where does it go when it is not present? It is certainly not in the body otherwise we would be able to see it all the time. Logic is telling us here and mind can disassociate from body and senses, thus proving it is a distinct and particular entity separate from the body. These are the formal arguments Nyaya give for the existence of the soul as separate from the body. Now where in this line of reasoning is a single appeal made to testimony?
Samkhya go further than Nyaya because Nyaya deals with only particulars, but Samkhya is more interested the cosmology of how particulars arise through reducing the Nyaya categories even further into just two irreducible categories of observers and objects in to prove purusha and prakriti. Again they do this using formal logical arguments and not a line from testimony is cited.