Champernowne's construction (and some that I have invented, too) generate all possible finite bit patterns with the proper frequency. These sequences are not random, but they do satisfy the strong law of large numbers. The frequency of 0 and 1 is 1/2; the frequency of 00,01,10, and 11 is 1/4,; etc.
These sequences can be combined with pseudo-random-number generators to break up obvious correlations in the geneators. It's useful, but still doesn't produce a random sequence.
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.