I am not claiming that Champernowne's constant is a random number! I am saying because it is not a random number, Omega cannot be in it - and neither can Shakespeare.
Then you are denying Champernowne's (and others) proof. Mathematics (and scientific endeavors in general) are not like à la carte menus, you have to take the whole thing, you cannot pick and choose which theorems you with to accept. Once the axioms are agreed, the consequence are not in dispute.