I had been watching a video on philosophy of mind where one of the speakers says that, in the past, people thought that humans were very complex machines: some sort of automaton theory. Everything we did and how we acted could allegedly be explained by purely mechanistic processes, which I imagine would be things like fluid pressure and muscles acting like pulleys to control our legs. They compared it to how the computational theory of mind has been dying and that it is interesting to see how we have shifted our view about how humans work to our most advanced creations. Similarly to how it was naive to think that a man is not significantly different from a series of gears and pressure valves, so too was it naive to think that the computer is a good analog for the brain.
While I think the parallel is quite nice with the slow death / evolution of the computational theory of mind, was something akin to the described "automaton theory" ever commonly held? It certainly seems reasonable to think that a bunch of DaVincis could create an artificial man out of cogs and Archimedes screws, but I suppose that doesn't mean anyone really thought it was true.
Commonly, no, but there were people with this view. Descartes introduced the idea that space is completely filled with particles moving in vortices (in Traité du monde et de la lumière). Everything happened by these particles pushing on one another - there is no "action at a distance" (and in particular gravity and light are actions in this medium, not something passing through a vacuum). He though of the actions which transferred thought to action as being mechanical. However he was a dualist, and held that the mind or soul was immaterial and that it interacted with the body via the pineal gland.
La Metrie took this a stage further in L'homme Machine, saying that even the mind was corporeal and mechanical, which I think corresponds to your question.