Edit: This answer is rewritten due to extensive editing of the original question.
Your question: Does this reasoning have logical flaws that you'd point out ?
we will explain the easy problems, associated to overt behaviour (a public property) and it's (sic) cause in neurological mechanisms.
This sentence is not clear. Are the easy problems having a cause in neurological mechanisms, or is the overt behavior having a cause in neurological mechanisms? Either suggestion seems to reverse cause and effect. Since no explanation is forthcoming, I assume that 'we will explain' refers to a hitherto unmentioned philosophical program.
Note that we can't ever know someone's exact inner experience. But that's how we carry out science anyways.
This is a non sequitur. The scientific program is not carried out on the basis of scientists' exact inner experiences, unless these experiences are the subject of the research. When chemists perform experiments in spectrophotometry to establish the colour, they will measure the wavelength and these measurements are objective and replicable. That these measurements correlate with our subjective experiences is nice and helps scientists to communicate (cobalt blue means something), but the measurements could theoretically just as well be done by a philosophical zombie. The research would be indistinguishable and happily accepted and published.
When someone says "it's a blue sky", is it identical to our experience? This assumption of shared reality is just as present in standard scientific practice.
Yes, you can't prevent scientists from heaving subjective experiences. These experiences are, however, irrelevant for the outcome of scientific practice. If the outcome of a scientific experiment would depend on the subjective experience of the scientist (cobalt blue-ish) then it would be bad science.
we will trust those descriptions, and create a theory that explains the appearance and the subjective experience, as strongly emergent phenomenon or as a new kind of fundamental substance.
What you describe here is bad science. This sounds like auras, chakras and the like, phenomena that are firmly in the domain of pseudoscience. Strong emergence is controversial and in my opinion (yes, this is subjective, ha!) adjacent to magical thinking.
Summarising: inner experience exists, it is caused by neural activity,
Despite my critique on this discourse, I agree with this statement.
since it involves a new kind of substance,
An unsubstantiated and fact-free assumption that brings us straight back to unadulterated dualism.
Your first conclusion is that consciousness exists, but some (of us) could be zombies. This seems irrefutable and is in fact a well established idea since Descartes. Though, I don't recall Descartes mentioning zombies. This is more a reassertion of what was written above.
Your second conclusion is that consciousness likely emerges from a neural correlate of consiousness. Maybe. And it is ontologically not the same.
I take it that your third conclusion asserts that neural correlates of consciousness cause behaviour of which we are conscious and that this hopes to be an answer to the preambule: Does neural activity cause consciousness?
I don't think this necessarily follows. Mind you, I do think that consciousness is caused by neural activity, but it is called neural correlate of consciousness for a reason. Correlation is not causation. Neurological activity is detected in patients under general anaestesia. These patients, one hopes, are not conscious. In another answer, the documented navigation skills of rodents and humans were correlated with brain activity. The subjects (the humans at least) reported subjective experience. I would, however, like to draw attention to the phenomenon of sleepwalking. People who sleepwalk show navigation skills but seem to be unresponsive and have often no recollection of their whereabouts during their nocturnal sojourns. In these cases they do not report subjective experience. In fact, they are often very much surprised with their exploits. There are cases of people sleepwalking that involve the subject driving a car.
So far, or at least as of last year, it appears that the neural correlate of consciousnes has not yet been found.