I greatly appreciate this helpful and timely information. At this point I'm thinking about how best to construct a short list of available sources/suppliers that meet hardware specs. Doing that would probably be a valuable contribution to the TOS prop-making community of interest. First, what's the right hardware?
FURTHER RESEARCH: There's still the question that @DigiGal raised about apparent Burke 116 hardware variation. Knowing the right answer to this would resolve how to come up with a validated set of hardware specifications that work in every instance that one might encounter upon attaining this type of chair from its original production runs. At this time, all we know is that @DigiGal's Burke 116 hardware differed in specifications from mine, and that raises questions. I'm open to exploring why that is, and all I know to do is reverse engineer and use a process of elimination along the way during this build. I can't tell you why just yet there was this difference, but I can determine what it's not to a point. I know that it's not a flat shank machine screw because in both instances, I can say with confidence that it's countersunk, and chamfered at some angle in degrees that is consistent across the board. Researching this question to ground is going to be challenging, and have reason to think the designer is still alive after looking around on YouTube.
To break it down in this way for each hardware item would allow for the development of Burke 116 "TOS chair hardware packages" (unless someone has already done this). Having Burke 115 or 116 hardware packages -- and some upgrade options on top of that -- would be valuable to many interested parties. These envisioned hardware packages would come with a variety of options because most of the hardware on a Burke 116 isn't visible once assembled. The only hardware visible on Burke 116s is partial in the four countersunk machine screws that go through the propeller's pedestal mating flange into the bottom of the seat bucket, which only showed up in a few scenes where the chairs were knocked over like what @DigiGal isolated from a survey of all regular TOS production stills. In addition, only a portion of these four machine screws are visible when assembled and are being painted over during the process of replicating this particular ST prop. Given that, the rest of the hardware that makes up a Burke 116 chair is left to the imagination so long as it works, doesn't reduce the chair's structural integrity, or alter the functionality of the prop such as inhibiting the memory return feature.
One conclusion is that it's fine to improve upon the material and appearance of Burke 116 hardware sets or individual items if so desired by the prop-maker. The entire hardware package could be color anodized aluminum alloy, or even 24 carat gold-plated titanium for that matter, but probably more along the lines of http://www.fastener-express.com/flat-head-socket-screws-red-anodized-aluminum.aspx
The general conclusion about the question of Burke 115 or 116 hardware quality is that only the complete hardware specifications themselves act as the Burke 116 cosmological constant, ceteras paribus. We are not certain the scope of this yet. To me, it's worth the trouble to verify because once this becomes a known, you will have a common denominator and there's much less angst facing prop-makers, vendors, consumers, 3D printer operators, etc, etc. I assume either rightly or wrongly that there is also an unsatisfied demand for these chairs out there that could be tapped into on a grand scale, general public consumer level.