Cantor's diagonal argument consists of two parts: bijection and the extraction of the new number. If he shows that a given architecture of bijection doesn't work, why does it imply that any other architecture of bijection should not work either?
Just an addendum to make my point clearer:
Another possible architecture that comes to my mind is to write real numbers in the table and correspond them to the nth power of 2. Then we start to construct new numbers that weren't listed in our table and correspond them with powers of 3. After that we construct new numbers from the table of power of 3 (even if without checking we had them in the first table) we correspond them to the nth power of 5, and so on and so forth. It is easy to notice that this is a poor bijection architecture as you can put them back into one list and just repeat the construction of the new number. Why can it be mapped back to a simple list for any architecture?
On the other hand we can easily think of correspondence of natural numbers to itself, in a way that we will end up having extra unlisted numbers.E.g.(1->2,2->4, etc...). Obviously it doesn't imply that there are more natural numbers than 'natural numbers'.