Cantor's Diagonal Argument Why Cantor's Diagonal Argument to prove that real number set is not countable, cannot be applied to natural numbers? Indeed, if we cancel the "0." in the proof, the list contains all natural numbers, and the argument can be applied to this set.

How about this slightly different (but equivalent) form of the proof?  I assume that you already agree that the natural numbers $\mathbb{N}$ are countable, and your question is with the real numbers $\mathbb{R}$.
Theorem: Let $S$ be any countable set of real numbers.  Then there exists a real number $x$ that is not in $S$.
Proof: Cantor's Diagonal argument.  Note that in this version, the proof is no longer by contradiction, you just construct an $x$ not in $S$.
Corollary: The real numbers $\mathbb{R}$ are uncountable.
Proof: The set $\mathbb{R}$ contains every real number as a member by definition.  By the contrapositive  of our Theorem, $\mathbb{R}$ cannot be countable.
Note that this formulation will not work for $\mathbb{N}$ because $\mathbb{N}$ is countable and contains all natural numbers, and thus would be an instant counterexample for the hypothesis of the natural number version of our Theorem.
