I am confused. Isn't this exactly what I am proving? That the functions / results DO form uniformly, over x sized range, within y amount of keys?

By you saying, "...actually proving that cryptographic and hashing functions don’t generate uniformly distributed numbers", that means you believe the hashes do generate uniformly numbers, correct? If so, what is so hard to understand lol?!
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The uniformity refers to the equal probability for any value to show up at any time, not to spreading around N values over some X space. That one isn't uniform at all lol - it's binomial, e.g. the consequence of a uniform distribution.... that's why you get repeats, tripeats, 100-peats and so on, with lower or higher degrees of certainty.

However none of this should allow any kind of prediction, no matter how many values you use as some basis to build up some prediction model. Even if you have the entire set of the distribution repeated millions or quadrillions of times each element, you only have 2 possible non-normal scenarios at every given moment:

1. You observe some values don't appear as much as expected, and the probability of this event happening after so many trials is basically zero (not 0.01, but more like 1/infinite).
2. You observe some values show up way too many times, and the probability of this event happening is also basically zero.

At this point you can say that the thing is either rigged, or that you were born really unlucky. You might as well claim we're leaving in a simulated reality at that point.

Any other state lies every possible value's frequency probability anywhere between +0% and 100%, and anything can happen, pretty much.

That's why people will say that trying to find order in chaos is exactly what "breaking cryptography" is defined as.

Did you do it?