Yes, I'm aware of that, but being irrational alone is not sufficient. There are an infinite number of irrational base-10 numbers that only contain combinations of 0 and 1, for example, and none of them will contain my phone number, credit card, etc.
Not all irrational numbers are normal numbers, and only normal numbers are guaranteed to behave as described in the OP.

According to my mathematician wife, "it's got the same problem most paradoxes do in that it's self-referential".