I think the finite information content comes from being an element of a countable set. Like every other real number, the digits of Chaitin's constant themselves form a countable set (a sequence), while that set is a member of the uncountable R. Similarly, the busy beaver set is a subset of N, and drawn from the uncountable set 2^N.

Countable sets are useful (or rather, uncountable ones are inconvenient) because you can set up a normalized probability distribution over their contents. But... the set {Chaitin's Constant} is countable (it has one element) but I still can't get Omega's digits. So there still seems to be a bit of mystery here.