I don't think most computer science students learn even a fraction of this intellectual history.
Judging from my personal experience with engineering students over 37 years, I doubt that many computer science students have ever heard of Boole or Frege as philosophers or read any of their works, although they might be familiar with Boolean logic. The stereotype of engineers as philistines, like many stereotypes, is pretty much true.
These various old time computers such as the Babbage machine did work but did not have memory. Memory is key to the modern computer and sets the modern device apart from the Babbage machine.
Babbage's design for the Analytical Engine did in fact include memory: 1000 numbers of 40 digits each. The Analytical Engine was never constructed because it was too mechanically complex to be built at any reasonable cost using the machining technology of the time.