I appreciate your lengthy explanation and I largely agree with it, even though it probably doesn't help much because anybody who has not understood this yet will have stopped reading latest at 20% in. That's not your fault but just based on the observation that attention spans are short and people strongly prefer to spend their time on reading things they are interested in. And anybody interested in this subject who has this much time to spare has likely already done that elsewhere. But I'd be happy to be wrong and would welcome if just a single person gained better understanding through your text.
What I would suggest though is to avoid using the term "random" for this. I know you put it in quotes, but that term is so over-misused that we do it a disservice by adding more misuse. Numbers (or other objects) are not random, it's the process that produced them in some context that's random. Random processes have very specific, mathematically well-defined properties, just like the concept of decidability has, and throwing around the term with other meanings really does it a disservice, similar to doing the same for decidability.
What you are probably looking for is a term like "arbitrary".
Sure, “arbitrary” is a suitable term here too. A random process is just one way to generate an arbitrary function.
My use of “random” here was referring to the coin flipping process and was more for building intuition than precisely specifying all the other non-expressible functions. I was trying to allude to the fact that these other type of functions don’t have any easily expressible process behind them. When I’ve taught this stuff in the past I’ve found that people latch onto coin flipping more easily that imagining some arbitrary assignment of values.
For what it’s worth, I used to be a researcher in probability and information theory and have published papers on those topics so I am aware of the various technical definitions of randomness (Kolmogorov axioms, algorithmic probability theory, etc.)
I think you’re right about my comment being a little too lengthy for most people to find useful. I started explaining this stuff and got carried away.