- Maybe statistics and the HUP are conjoined in an inseparable way in QM, *
Yes, they are, because a quantum system has the property that some observables are non-defined. The standard QM says that this property is not a flaw in our measurements, but an intrinsic property of the quantum system itself.
Let $x$ and $p_x$ be the two non-commuting operators. If you want to measure $x$, and the wave-function according to which you prepare particles is not that of a well-determined position, then about each particle and particle that you prepare, you cannot say before the measurement that it has a position. So, a measurement on one single particle collapses the w.f. on $\delta (x - x_0)$ where $x_0$ is one of the positions allowed by the initial w.f. Therefore a measurement on one single particle says nothing. For assessing the values allowed by your w.f. you need to measure many particles
Next, after measuring $x$ on one particle you destroy the of initial w.f. s.t. on the measured particle it makes no sense to measure $p_x$. This is why you need an ensemble of particles prepared according to the same w.f., so as to measure on part of them $x$ and on the other part $p_x$.
So, in order to check the uncertainty principle, you need many particles identically prepared, and that because of the quantum nature of the particles.
but if say, all operators could commute, there would be no HUP. You need the ingredient of non-commutativity which has nothing to do with statistics. Is that correct?
If all the operators commute we have classical mechanics. One of the proofs of the HUP is based on the fact that the commutator is non-zero.
Now, the non-commutativity is indirectly connected with the statistics. The fact that the two operators don't commute, implies that at least one of them is undetermined, has no fixed value $\to$ statistics.
A simple example: the oerators $x$ and $\hat p_x = m \hat v_x$ don't commute. Let's fix one of these observables, e.g. $x=x_0$. What can we say about the velocity of the quantum particle? If we believe to QM, $v_x$ becomes completely undetermined. Let's check that by measurement. How do we measure velocity? Some time $Δt$ after fixing $x$, we measure $x$ again. Let's say that we find $x=x_1$. So, we calculate $v_1 =(x_1−x_0)/Δt$. But, let's check this result on another particle, that we also fixed initially at $x_0$. After the same interval $Δt$ we measure the position and find, for instance, $x_2 \ne x_1$. So, we get $v_2 \ne v_1$. Then, to which one of the results to believe, to $v_1$, or to $v_2$? No way to prefer one of them. Then, maybe QM is right and for fix $x$, the observable $v_x$ is undetermined?