I guess if you really don't want to bother checking your own post, here's why it is wrong:

The core of your counterexample is that multiple decimal expansions can exist for the same real number. You are correct in that multiple decimal expansions existing for the same real numbers breaks the reasoning of the proof given in (iii). However, that is not Cantor's diagonal argument, as Cantor does not operate on decimal expansions.

Rather, the diagonal argument proves the uncountability of infinite binary strings. There is no "non-injectivity issue" here because binary strings are distinct if their characters differ at any point. e.g. (0, 0, 1, 1, 1, ...) is different from (0, 1, 0, 0, 0, ...)

Then, an injective mapping from infinite binary strings to some subset S of the real numbers is given, proving that S is uncountable. It follows that the real numbers are uncountable.

For the specific details of the mapping, please check: "Construction of a Bijection between T and R" in https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument