First off, your 'The end result' quote does not appear on this page, so the inaccuracy of that statement doesn't really seem to me a defect of the page.
But backing up, I do not see how cause and effect are mixed up - the relationship between position and momentum is the Fourier transform, as the post says, and that is why they have a nonzero Poisson bracket, and that is why they don't commute. Yes, it only speaks of the position-momentum case, and it doesn't consider more than one dimension, and it leaves out a highly technical middle step. At this level of discussion, I seriously doubt that going further is fruitful. This gets across an important point. That there is way more that could be said does not weaken it in the least.
And... 'plenty'? You gave the spin operators as an example. What's another? Energy-time, if you're working in QFT and you measure some spacetime region selected out by some other operator, against a timelike-oriented tensor... and in that case, a very similar Fourier transform reasoning applies.
At this level of discussion, I seriously doubt that going further is fruitful. This gets across an important point.
Unfortunately, it also gives an illusion of understanding, since it does not explain that this is only one example out of many (why do different components of momentum commute, but different components of angular momentum do not?). As for the Fourier transform, arguably this can be presented as a result of position and momentum being conjugate variables/canonical coordinates. Speaking of which, any pair of classical canonical variables probably translates into a version of uncertainty relation, even such unusual ones as action/angle, fluxes of electric and magnetic fields, etc.
Today's post, The So-Called Heisenberg Uncertainty Principle was originally published on 23 April 2008. A summary (taken from the LW wiki):
Discuss the post here (rather than in the comments to the original post).
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