[vgatherps]

> The proof of uncountability of real numbers starts with a process of enumerating this infinite list of infinitely long binary numbers.

> How could the diagonal construction ever finish to allow a "check" of whether the result of that process is actually on the list.

> The diagonal just seems like a shortcut to a number that we know the first process hasn't gotten to yet?

I think you are mixing up the enumeration of the (hypothetically countable) reals with the existance of said set of reals, and viewing the construction of the counterexample as an actual process that involves programmatically iterating over the list instead of a description of how the digits of said new real number is created.

There is no pause or setp-by-step of this iteration that's required, since it's only telling you that the i'th digit of the counterexample will be distinct from the i'th digit of the i'th number (thereby creating a real not in your countable set, since you have created a number outside of your 1-1 mapping onto the natural numbers).