Dow and Tall's result doesn't use a supercompact. The supercompact was used in Nyikos' earlier result.

Then it should be possible to describe your universe of random variables via one giant Ord-length directed system, where at stage $\alpha$ we add in all random variables of set rank $\le \alpha$ of all possible dependencies on previously constructed random variables. I doubt this will satisfy your intuition any more than my first comment did, but I really don't think any successful approach to your question will differ significantly from what I've described.

In your vaguely conceived theory of random variables, how do you propose we avoid the following Russell-like paradox: for any class of random variables $V,$ we can define a fair coin flip independent from each variable $A \in V.$ My point is, it's not clear we can even develop a theory of arbitrary coin flips, let alone arbitrary random variables.

For your example, can't you represent $f$ through an Ord-length directed system of approximations, i.e. $\langle (f \restriction \alpha, \Omega_{\alpha}), \alpha \in \text{Ord} \rangle?$

@Gerry As David says, it's Singular Cardinals Hypothesis.

There's also Joel's answer here mathoverflow.net/a/270540/109573

See mathoverflow.net/questions/269682/…

A locally recurrent function might take every value in its range an uncountable number of times (e.g., if you compose an arbitrary locally recurrent function with the projection of some space-filling curve). So this doesn't necessarily meet the requirements of the question.