It's important to distinguish between Babbage's Difference Engine and Analytical Engine. The Difference Engine is essentially a specialized mechanical calculator for producing pre-computed tables, but was not completed at the time due to the high cost of manufacturing (a working No. 2 Difference Engine was built between 1989 and 1991). It had some innovative features and was fairly early, but wasn't especially important within the history of mechanical calculators. The Analytical Engine is the interesting one. It was definitely intended to be programmable using punch cards and have the ability to read and write data from and to punch cards. People who have spent time reading Babbage's notes argue that it would have been Turing-complete, but it would also have been extremely difficult to program. However, after the failure of the Difference Engine project, Babbage was unable to get funding to build more than a fraction of his Analytical Engine design. As a result, it was basically forgotten until the 1940s, and is only mentioned as a historical curiosity rather than a concrete source of inspiration in that period. Similarly, the early work on formal models of computation (mostly in the 1930s) does not reference the Analytical Engine at all, even though it would have been as powerful as the Turing machine. Historians in the 1920s might have been aware of either or both designs if they were focused on the history of mathematics or calculating machines, but it would have been as an early example of a calculating machine that was never completed, not as something distinctly more powerful, which the Analytical Engine would have been if it were ever finished.