Is the set of all possible finite alphabets uncountable?
Your approach can essentially be summarized as:


I define an injection between $\mathbb{N}$ and my set $S$.
This injection is not a bijection because some elements of the codomain are left unmapped.
Therefore there exists no bijection between $\mathbb{N}$ and $S$.


Stated like this, it is perhaps clear that the last statement does not follow from the others. An example of where this reasoning would fail is if we let $S = \mathbb{Z}$ and let the injection be the identity function (mapping naturals to themselves). This leaves all the negative integers in the codomain unmapped, but we know that $\mathbb{Z}$ is countable.