Interesting question. I can see where Buckethead is coming from. I too, tend to draw a distinction between the inability to measure something precisely on one hand and a total randomness (as in lack of cause) on the other.
For example, classical mechanics is considered fully deterministic. In theory if you know the positions and velocities of all molecules in 1 litre volume of gas with sufficient accuracy, you should be able to predict their positions at some future time, say 1 second ahead. In practice, something like moving 1kg of mass by 10cm somewhere in the vicinity of Sirius is sufficient to throw spanner into the works here on Earth and make the positions completely unpredictable (Penrose gave this argument in one of his books). Then there are pesky questions about knowing the position of all particles in the universe, solving gazillion-body problem, not to mention photons coming from the very fringes of the observable universe which you can't predict because you haven't seen them yet.
Nevertheless we still call it deterministic. Why is that? I think this is because, when a (classical) particle hits the screen, we don't just say "it's random", we have an explanation ready, we say, well it hit here and not there because the sum total of all forces must have been such as to produce this kind of trajectory. If only we knew the forces beforehand we could've surely predicted where it was going to hit.
Now let's look at HUP. We know what it says but why exactly does it say it, where does it come from? Well, from non-commuting projection operators. And where do these come from? From the observables, measurement, wavefunction collapse, Born rule etc. And these? At this point we are supposed to shut up and calculate.
But but but. Just like with the gas pressure, to get more and more accurate results we will have to look at individual molecules, so with quantum measurement we will have to treat the entire measurement apparatus quantum-mechanically. Obviously we can't just replace a hugely complicated system with lots of interacting degrees of freedom (measurement apparatus) with a simple operator and expect to get exactly the same results? Surely this must be some kind of idealization, simplification or generalization just like the gas pressure is the generalization of the forces of individual molecules hitting the wall?
What I'm driving at is, the process of measurement, wavefunction collapse, observables, their projection operators, and therefore HUP are all likely to be emergent phenomena. In other words, HUP is valid for a ideal measurement which is only an approximation for the real measurement, arising from our ignorance of quantum-mechanical nature of the measurement apparatus and its environment.
So, when (this time quantum) particle hits the screen we could say, well it hit here and not there because the relative phases of the wavefunctions of everything the particle had ever interacted with (the atoms of the screen, the source, the two slits, the CMBR photon that was passing by and everything that was entangled with it since the beginning of time), yeah all these phases just happened to be aligned so. Yeah, if only we knew all these phases beforehand we surely could have predicted where it was going to hit. Honestly
