If you take number generator and get a different random result every time you call it, you have no idea whether it is deterministic (algorithmic) or not. One can try to do statistical tests, but these can reveal only weak algorithms, not the sufficiently good ones.

> Further, if quantum states are described by wave functions then they must be intrinsically random due to the mathematics of the Fourier transform.

That is a very strange statement. Which part of mathematics of the Fourier transform suggest wave functions are intrinsically random?

The mathematics of the Fourier transform enforces that the frequency spread times the time spread is equal to a constant. Therefore if the quantum state of a system is fully described by its wavefunction, then the system absolutely cannot have predetermined values of conjugate variables. It takes many more wavenumbers to describe a wavefunction well localised in space.

Big if. Real systems interact with their environment, which means every psi calculation is approximate, as there is no limit to the possible expansion of domain the wave function is defined on. If we want full description, we end up with the necessity to work with wave function of the whole universe. This means that argument for non-existence of simultaneous values of conjugates does not work. This was after all analyzed by EPR and Bell and the result is, conjugate pairs may simultaneously exist in non-local theories.

This is the core of your disagreement. You suppose Occam's razor dictates something appearing unfalsifiably random is in-fact random. The other side supposes that you're the one making a stretch, since everything which can be falsified categorically turns out to be deterministic.

I've never heard of this. Why do you think this is true? Is that because QM is our first example of inherent randomness in physics and all previous physics was deterministic? I don't understand the argument you are pointing out

No, it's a purely logical argument. Randomness is fundamentally unprovable/unfalsifiable. Therefore, in principle, any possible process in any imaginable system can only ultimately be found to be either provably deterministic or unfalsifiable.

On the other hand, it is possible for there to be unfalsifiable structures which determine the outcome of apparently random processes which are not truly random. In fact, it's trivial to create practical cases of such sources of psudorandomness.

You believe that positing the existence of one such structure to be the bigger leap simply because it may be not just practically unfalsifiable but in-principle unfalsifiable (though that is undetermined).