If world did run on real numbers that we could harness for computation I would be more than happy, because using those we would be able to perform hypercomputation. See https://en.m.wikipedia.org/wiki/Real_computation

However this is forbidden by Bekensteins bound, so unless modern physics is horribly broken it’s ruled out at least in any sense visible to us even in principle.

Not a quantum physicists, but IMO Bekenstein bound is not applicable here, because quantum laws are non-deterministic, therefore you can describe the structure of a system, but you cannot describe how it will evolve. Quantum randomness might be in the very essence of how the brain and mind works.

Quantum randomness being necessary hardly seems like it would have profound practical implications since augmenting a digital computer with a geiger counter would be trivial.

It's much easier than that. Hardware random number generators are often based on something like a reverse-biased diode. Electrons migrate across and it's entirely random when it happens. Amplify and count them and you get a great source of entropy.

DNA and proteins are obviously discrete, so even if the hardware relies on some fundamental analogous behavior, the 'software' and each hardware component can still be analyzed as discrete behavior.

However, for anything operating at human temperatures we can reasonably assume that any effective behavior can be simulated by discrete operations, as any nuances of fundamental analogousness would be drowned by thermal noise and the amount of precision that any behavior can require is rather low, much lower than e.g. any standard floating point number in a discrete CPU.

I think the assumption that software may only be digital is the limited one.

Otherwise it becomes a meaningless, all-encompassing term.