Continuum Hypothesis - Why does my "proof" fail?

Your underlying assumption that "everything can be indexed nicely" basically translates to the following:

For every infinite set $X$, there is some set $Y$ such that $X\equiv (\aleph_0)^Y$, that is, the set of "$Y$-tuples" of natural numbers.

This, however, is provably false in ZFC! It turns out that we can show in ZFC using Koenig's Theorem that $(\aleph_0)^Y$ can never have size exactly $\aleph_\omega$.

In general, when reasoning about infinities, you need to be very careful and precise. Informal arguments like what you've written above - which hinge on appeals to intuition (how do you justify that every set can be "indexed" nicely?) - are more likely to confuse you than to help, until you have learned enough set theory to know when to trust them.

Incidentally, it's worth pointing out that we know that ZFC (assuming it's consistent!) can't prove the Continuum Hypothesis: if you give me a model of ZFC, I can produce (via forcing) a model of ZFC in which the Continuum Hypothesis is false (and another in which it is true).

You are right that for any uncountable set, if you try to index it with tuples of natural numbers, you will need at least $\aleph_0$ indices, so your set of all possible tuples you can create has cardinality $\aleph_0^{\aleph_0}=\mathfrak{c}$. But what if you don't actually need every possible tuple to index the elements of your set? That is, maybe if you try to label all the elements of your set with $\aleph_0$-tuples of natural numbers, you can do so, but no matter how you do it, you will have some $\aleph_0$-tuples left over at the end which you haven't used. This would mean that your set actually has cardinality smaller than $\mathfrak{c}$. And yet the set still might be uncountable: just because you can't label all the elements with finite tuples, doesn't necessarily mean that if you use infinite tuples, you will need to use all of the possible tuples.