Imho it was always "computerized", they just didn't have a computer. To me the approaches used in the early 20th century look like people defining a simple VM then writing programs that "execute" on that VM.
Exactly. The step to formalize mathematics through computation is just the logical consequence of the program of the formalizers.
The idea actually goes back to Leibnitz, who was very much overoptimistic about computability, but already conceived of the idea of a logic machine, which could deter the truth value of any statement.
> Imho it was always "computerized", they just didn't have a computer.
They had a whole lot of computers, actually. But back then the "computers" were actual people whose job was to do computations with pen and paper (and a few very primitive machines).
