The name of the "uncertainty" principle is pretty stupid and it has nothing to do with its meaning.
The so-called "uncertainty" is just a trivial property of the Fourier transform. It is the same property which requires a great frequency bandwidth for transmitting a pulse that is short in time.
The fact that certain quantities that appear in quantum mechanics are related by Fourier transforms is something that is completely orthogonal to the interpretation of the wave function amplitudes as probabilities of intrinsically random phenomena or as interaction efficiencies caused by real angles of rotation of the interacting particles/waves, which are unknown due to unknown initial conditions.
Technically you're correct, but you seem to be shortchanging the import of the uncertainty principle being a consequence of the Fourier transform.
That "trivial" property of the Fourier transform fundamentally limits our ability to measure both position and momentum, or time and energy of a particle. Without that aspect of QM we could get effectively unlimited certainty of those various properties. It deeply implies the universe is non-local in nature.
What is the difference between "unknown (and unknowable) initial conditions" and "uncertainty"? You can't go back in time to check the initial conditions.
The uncertainty in the initial conditions has no relationship with the "uncertainty" word used in the phrase "uncertainty principle", which refers to the relationship between the variances of two functions, one of which is the Fourier transform of the other.
I have refrained from using a phrase like "uncertainty in the initial conditions" and I have said "unknown initial conditions", precisely to not imply any connection with "the uncertainty principle", because no such connection exists.
Moreover, "uncertainty" is typically used about the difference between the true value of a physical quantity and its estimated value, but in quantum mechanics problems there are many cases when a value is completely unknown (i.e. all the possible values are equi-probable at the initial time, like the angular coordinate of the position of an electron bound in an atom) and not only uncertain.
Saying "unknown initial conditions" covers such cases.
Max Bohr explained the uncertainty principle in his Nobel laureate speech quite well (and in nontechnical terms) - even though it might be interpreted as some law today, it definitely began as a practical obstacle.
Also, the hidden variable theories seem to be about practicality; for all intents and purposes, because it would be practically impossible to know that such hidden variables exist, they are deemed as non-existent.
> Also, the hidden variable theories seem to be about practicality; for all intents and purposes, because it would be practically impossible to know that such hidden variables exist, they are deemed as non-existent.
Definitely not. Everyone would strongly prefer a model of the universe that was deterministic. Hidden variables don't get ignored just because we don't need them to make predictions. We really really really want them, for their own sake, but we cannot make them work.
I think you are folding two things together that are related but not the same. Hidden variables are about the state of an entangled system. The uncertainty principle can be applied to measurements taken of an entangled system but can also just refer to two quantities of a single particle such as it's position and momentum.
