Why do we consider enumeration up to $w$ instead of leaving it to as many ordinal numbers?

A few minutes ago I asked a question about a "proof" that $ \mathbb{R}$ is enumerable that crossed my mind: What's wrong with this "proof" that $ \mathbb{R}$ is enumerable?

I was told to look into ordinal numbers, and that after crossing $ \omega$ we stop considering something to be an enumeration.

Why is this the case? Are there negative consequences if we don’t put this limitation?

Edit: I always thought of $ \mathbb{N}$ as the "counting numbers" – but… when we cross over to ordinals like $ \omega$ , $ \omega+1$ , etc, aren’t we still effectively counting?