If you know the proof you probably know that you don't actually use the "(n+1) mod 10" mapping, you use that mapping but instead remap all 8's into something else, like 3, in particular you don't remap 8's into 9's to avoid the "99999..." tail problem. In fact if you like you can just use the following mapping: 0->1, {1,2,3,4,5,6,7,8,9}->0. Also, for the starting representations you simply replace all expansions with an infinite tail of "9999...", as for every such representation there exists a representation that does not end in "9999...". This makes the decimal expansions unique.
All the name-able reals are obviously countable, this has a deep connection with the Löwenheim–Skolem theorem: https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...
1: 1
2: 2
3: 3
4: 4
5: 5
...
But maybe your point about needing infinite digits is the reason? Anyway, here's a sketch of the "proof":
list the natural numbers, in binary, simply in order, from zero onwards. But written in reverse, eg 100 (4) is written 001
[NB: the list will grow longer faster than the length of each number, so the diagonal will be all zeros.]
now construct the "diagonal" number: taking its ith digit as the complement of the ith digit of the ith number in the list. This will be all 1's.
This number isn't in the list, because one of its digits differs from each one of them, by definition. Therefore, this list of the natural numbers can't enumerate all the natural numbers.
Perhaps the flaw is simply that although there are infinitely many natural numbers, each of them only has finite digits. That all-1 diagonal has infinite digits, so it isn't a natural number (as you say).
There already exists at least one mapping of integers into integers, the identity mapping. Hence a correct proof that there exists no mapping of integers into integers (proving the opposite claim) would also tell us that FOL+ZFC (first order logic with the standard axioms of set theory) is inconsistent, regardless of whether Cantor's diagonal argument features in that proof.
In the other direction, if FOL+ZFC is inconsistent, it's trivial to write a proof that shows that integers cannot be mapped to integers that features Cantor's diagonal argument in it.
So the existence of a proof that shows that no mapping of the integers to the integers exists is equivalent to FOL+ZFC being inconsistent, and no-one has yet found FOL+ZFC to be inconsistent, or expects this to be the case.
If a proof technique could prove untrue theorems, it wouldn't be very convincing, would it?
I hope someone comes here to argue that Vitalik is right against me.
In the second paragraph, there is a faulty assumption that all real numbers are computable.