First: I appreciate your pressing your points, as it's helpful for me to clarify my own reasoning. This hinges on a few points, and again the sense of ideas as interfaces / models is key for me. I'm less a believer in
truth than of
pragmatism of ideas, something I've been delving into through history of philosophy for a number of years now. And it's the
utility of dividing systems into analogue vs. digital which seems key.
And again, this allows for differences and distinctions, one of which is your view, which again I find less useful and clear. That is, crystal-driven timepieces strike me as more usefully considered as digital rather than analogue.
On the distinction between a mechanically-implemented logic vs. an electronically implemented one: the degree is largely of complexity, scale, and speed, but in general once you've ventured into the electronic domain, it becomes infeasible to provide a comparable utility or function by mechanical means. We've tried mechanical digital calculators and computers. We've abandoned them on account of cost, slowness, unreliability, size, and power consumption. Mechanical systems cost too much, ran too slowly, broke down too often, and simply could not scale in the way that semiconductor-based systems could.
To the extent that a mechanical function (say, a mulitiplier gear) represents a pure logical function, we can abstract that functioning from the gear itself, much as writing conveys meaning independent of medium.
It's less possible to divorce the mechanical functioning of such a system from its inherent parts: their materials, mass, size, and the like. If we look at purely mechanical systems, they're very tightly linked to the inherent material constituents in a way that pure digital logic isn't. Let me give two examples.
The printing press (as a mechanical system, I'm not arguing its digital attributes if any) saw a profound development over the course of the 19th century. At the beginning of same, it was little evolved from Gutenberg's early adapted wine press, and with skilled operators might produce ~120 impressions an hour, a sheet every 30 seconds. Converting from a wood to a cast-iron frame roughly doubled that. Further developments: electrical power, cylindrical plates, web-based paper feed, increased that by the end of the century to one million impressions per hour, four orders of magnitude faster. That is, the function and capability of the machine was intrinsically bounded by materials (and power sources and paper characterstics, etc., etc.) from which it was constructed. Taking this further, modern Web servers / application servers are capable of millions of requests per second, another three orders of magnitude faster.
(In general, only one or two orders of magnitude is a fundamental revolution in capabilities: walking (5 kph) to bicycle (32 kph) to automobile (130 kph) to jet airliner (1000 kph) are separated by roughly 1.5 -- 2.5 orders of magnitude, with the largest step (e^2.2) being between the first two.)
Of digital systems, the unrelatedness to fundamental materials is probably best exemplified by virtualisation. That is, we see tremendous adoption of entirely virtualised systems in which the basic logic functions occur entirely independently of the underlying hardware implementation. A digital watch can fundamentally be implemented entirely in software, in ways that I'll venture a hardware watch cannot be. Though here again we trip up on my ideas / language / models distinction: is modelling a hardware system in software the same as emulation? I'm ... going to stick with "no", Because Reasons, though I'll acknowledge the question, though a large part would be that a model is a simplification of the reference system, whilst emulation is a complete functional equivalence.
Similarly, we can run identical software on entirely different CPU architectures, command sets, and semiconductor substrates with few if any practical considerations. Operation is divorced from hardware.
Keep in mind that the function of a transistor itself is equivalent to that of a gate or valve: a small controlling input leads to a large controlled input, and indeed these are often called gates or valves in the field / historically. There have been mechanical and hydraulic computers of limited capabilities. But again, moving from mechanical to electronic components
I'm not arguing that gears aren't capable of logic. I am arguing that gear-based logics severely restrict the capabilities and increase the physicality and physical constraints in ways in ways which fundamentally differ from those of purely electronic systems, and are best considered "analogue".
There's also the point that gears ... rotate continuously, rather than discretely. We can modify that (e.g., with cams or similar designs), but there's still that continous rotational motion at core.