The repeatability is due to all the faces meshing at the same time. So 360 teeth vs 360 teeth. All teeth engage, every time.
Yes; I was proposing four faces, 360-against-360 and, separately, 361-against-361. The "idler" plate has 360 on one side and 361 on the other. A sketch would make it more clear.
The accuracy is due to the precision of cutting the teeth.
Quite right; there is also a smoothing effect because of many Hertzian contact areas acting in concert, which also increases stiffness. Smoothing is helpful because there's the hope of calibrating out the low-frequency errors.
The accuracy of the 3D printer is certainly subject to a number of variables. Resolution, at least, is adequate (11 microns on my printer). Classical CNC subtractive machining is another option, but it may not be quite as available to the casual builder. For mass production it would be the way to go, though, to make a mold for injection molding.
You could cascade them but it will be a mighty expensive sextant.
That's what's so cool about 3D printing---the parts are practically free. 20 grams of ABS for one of these plates will set you back one dollar.
I'll leave it up to you to calculate how many you would need to stack up to get to 0.1 arc minutes (6 seconds) steps, throughout a 120° range.
A stack of two stages (360*361) is needed to get to 10 arcseconds (and a 720-degree range, thus a reflecting circle, really, rather than a sextant). Three stages would allow a much larger pitch and increased resolution (the latter probably not needed).
The Hirth interface probably could be 3d printed, but I'm unsure if 3d printing can achieve the surface finishes required.
This is a potential issue and it might be where CNC could outdo FDM. One could maximize the potential of FDM, though, by using the most elementary FDM feature possible for the teeth: single plastic noodles from the extruder, diameter 0.4 mm. Those are quite smooth. Larger features made up of several noodles packed together: not so smooth (with layer heights of 100 microns, surface roughness is ~50 microns).
Additionally, the steps are discrete steps. There's no way to get a mid-step position.
That can be both a bug and a feature. It makes the sextant "digital" and contributes to ease of reading. On the other hand, fine interactive adjustments are likely to be less convenient. I don't know if that would take some getting used to: set the sextant, look, and wait, rather than simultaneously looking and sneaking up on the contact with the micrometer. (But with the index plates, no worries about micrometer periodic error or backlash!)
Thanks for the feedback. It would be fun to make a prototype; any suggestions from the list for sources of optics (front surface mirrors and shades)? I suppose I can just cannibalize a Davis Mark 3.