You don't need logic, set theory, Turing machines or any meta-mathematics to build practical software.
When I was ten, I had no idea about any of this stuff and yet when somebody showed me how to do arithmetic, variable assignments, comparisons and goto in QBasic (pretty much equivalent of Babbage's machine) I was able to write a simple drawing program and tic-tac-toe which checked whether one of the players won.
Add some IO and I would write a program which reads series of transactions and computes your bank account balance. Tell me what a matrix is and I would implement LAPACK for you.
You absolutely need those things for a great deal of modern programming. All modern programming? No, but a great deal.
Without that they would have been at a disadvantage. I don't see what is so hard about this concept to you.
Regardless, the simple historic fact remains that Babbage and Ada were both unable to build the machine, and unable to verify their suspicion that the machine was special, and unable to effectively communicate to their peers this suspicion. The prerequisite math for all three of these tasks did not yet exist.
When I was ten, I had no idea about any of this stuff and yet when somebody showed me how to do arithmetic, variable assignments, comparisons and goto in QBasic (pretty much equivalent of Babbage's machine) I was able to write a simple drawing program and tic-tac-toe which checked whether one of the players won.
Add some IO and I would write a program which reads series of transactions and computes your bank account balance. Tell me what a matrix is and I would implement LAPACK for you.