It's a bit suspicious that we can't name or describe any of the uncountable irrational numbers, isn't it? Not even a single example.

Cantor's proof is not constructive. It doesn't name an example. If you enumerated all irrationals based on the algorithm required to calculate them then the contradiction in Cantor's diagonalization proof just turns into a failure to terminate. But that suggests that there are fewer irrationals than integers, not more.