It is however something fun to do in a homelab outside of the normal learning and playing around, and that has merit for me. Hobbies are hobbies in part because of the interest, joy and appreciation they drive.
In another sense you're very wrong. The set of uncomputable numbers also has measure 1. So while there are lots of normal numbers Pi belongs to a pretty exclusive club of numbers we can actually look at.
It's not a particularly deep result but the fact that the set of computable numbers is countable will never not make me existential. Every number that we can actually express, that we will ever know, is no bigger than the whole numbers.
But in contrast to that, I'm also happy to grant the existence of constructions with countably infinite length when discussing theories outside of computability. So in that context, is it possible to generate an infinite string of random numbers? Well, it is so long as you believe you can generate any amount random numbers at all. Just repeat the process, whatever it is.
Is that a reasonable thing to believe? That's probably a philosophical question at this point. I'm certainly not equipped to answer it. But assuming it is possible allows for some interesting math, which I support for its own sake.
This is all sort of reminiscent of a mathematician friend's stance on the axiom of choice. If you were to tell them a vehicle is only guaranteed to not explode by the axiom of choice, they wouldn't use it. On the other hand, assuming the axiom of choice is true leads to some useful math, even if it's sort of sketchy ontologically. It's a lot like programming. Sometimes you can't fix the bugs in the underlying system and you just code around them to get your stuff working.
Actually I had something even stronger in mind, being able to pass the generated number around. For pi I can have a program that I can pass around and that will give everyone access to all the digits of pi, in base 16 we can even have random access to all the digits.
For generating a random Cauchy sequence things are not that easy. Locally I can just use a true random number generator - in case such a thing exists - and generate all the digits I am interested in on demand, storing them in case I have to look at a previously generated digit again. But I can not easily pass that number around, only the digits I have already generated. We would need a shared database where everyone can share all the generated digits.
Or I could try to replace the true random number generator with a pseudo random number generator, then I could just pass the seed around. Everyone would have to either agree how the sequence of random numbers is mapped to the sequence of digits or we would have to use a seekable random number generator. But this raises the question whether using a [specific] pseudo random number generator would still yield a normal number.
Let's say we knew we had a normal number, say we could prove pi is normal. Then the set of sequences defined by the decimal expansion of pi starting at the nth place for all n in the natural numbers would be uniformly random over all decimal expansions. I'm going this route because it in theory allows us to define each sequence by virtue of a single number rather than an infinite sequence of randint(0, 9). This I think would be our best case if it were possible. But you would have to select a number uniformly in the range [0, inf) which... you can't.
[1] https://en.wikipedia.org/wiki/Normal_number#Properties_and_e...
ok... click
> Pi may be a normal number
huh...
We do not have, and perhaps will never get, proof that Pi is normal.