OP here. The number and variety of special-purpose computing devices that existed before general purpose computers is astounding. The surprising (to me) conclusion is that the main impediment to the development of computers wasn't technology. After all, Babbage's machine could have been built in his time if funding hadn't run out.
Rather, the limitation was that people didn't have the abstractions, vocabulary, and mental tools to properly conceive of general purpose computers as a concept and to understand their usefulness. They couldn't see that devices as seemingly disparate as tide prediction machines[1], census tabulation machines, and loom controllers were all instances of a single, terrifyingly general idea.
From what I can tell, Babbage mostly understood this, but it was Ada Lovelace who grasped it fully. But her writings weren't understood in her time and had to be "rediscovered" a century later. For example, she wrote [2]:
Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.
This leads me to wonder: what abstractions are we missing today that will be obvious to future generations?
BTW I have a follow-up thread on the optical telegraph, a form of networking that long predates the Internet. [3] My long-term goal is to teach a course on computing/networking/information processing before computers, with a view to extracting lessons that are still applicable today.
[1] https://en.wikipedia.org/wiki/Tide-predicting_machine
[2] https://googleblog.blogspot.com/2012/12/honouring-computings...
[3] https://twitter.com/random_walker/status/1037031465735860224