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".