Stefan,
Hmm. you keep forgetting key bits so conclude wrongly. I can't blame you, I struggled for ages! But the video does derive this, or also see the end notes of my previous essay experiment using colours. Remember the x axis scale of the Bell curves is detector field angle, and there are TWO curves offset by 90o, so when one peaks & troughs the other is crossing zero (mid height), and vice versa. Now consider the finding not as 'spin up/down' but just as 'same/different' to the detector electron direction.
We'll say Bob and Alice can rotate their dials at will. When the SAME angle they get opposite results, when at OPPOSITE angles they get the SAME results. Now when they're at 90o relative to EACH OTHER (anywhere around the dial) the correlated results have maximum uncertainty. This is because; if the rate of clicks between Bobs two photomultiplier channels is say 60:40, then Alice's will be the inverse so CORRELATION ( a 'relative' function) will be at its poorest (50:50). Of course NEAR 90o it'll still be poorish, whereas near the SAME or OPPOSITE settings it'll be pretty certain. That reproduces the Cos curves of the interactions but this time SQUARED due to the value amplifications of the photomultiplier cascades.
In the colour experiment it was subjective. When asked how close red was to green it was NOT AT ALL (opposite) but then comparing red & orange, green & lime or buff & sand, in was CLOSE. So the distribution of 'sameness' was non linear. But now asked if yellow is closest to red or to green? (orthogonal switchover point ) Duh! the result was around 50:50.
That statistical non-linearity is tricky to get your head round at first then keep in there! But don't assume the correlations actually mean what many assume them to mean. The distribution is NOT the original particle momentum distribution itself but that simply repeats itself. The simple subjective essay experiment with a bunch of students gave a surprisingly good reproduction of the Bell curves from the correlations. (of course many just suggest, oh 'that's just proved QM'. Doh!)
Peter