Dear Declan Andrew Traill,
I am in full agreement with you that entanglement, "a nonlocal process inaccessible to the classical world", is a most serious problem facing those who wish a comprehensible universe. Like you, I find it possible to produce a classical model that violates Bell's theorem. In the following I will try to compare our two results, both of which lead to the 'impossible' result.
You propose an angle-dependent detection probability. If all hits are detected, then all hits count as +1 or -1 (in the QM theory). If certain hits are missed, this effectively lowers the 'average' reading (for that angle) to a number below +1 (or above -1). This lowering of the average value is effected by the cos(a.b) term.
The +1 and -1 come from Bell's very first statement defining the problem, and reflects the consensus interpretation of the quantum mechanics of spin as a half integral "nonclassical" phenomenon. Your essentially classical model seems to accept the QM interpretation of spin as a two-state entity, which is generally true from spin statistics and magnetic fields, but has never been proved or experimentally demonstrated for single spins in magnetic-field-free space.
In my classical model the 'hidden variable' is simply the 3D nature of spin which yields an angle-dependent deflection that matches the Stern-Gerlach data which has the well-known 'lip' pattern. My Stern-Gerlach-based model assumes 'perfect' detection since none of the atoms are lost; all atoms reach the target. But the registered spin component is less than +1, dependent on the initial angle the spin makes with the magnetic field. This yields exactly the cos(a.b) curve that Bell claims is impossible to achieve classically. It's only impossible when one forces all projections of atoms through the Stern-Gerlach apparatus to be maximum or minimum. Of course, the data shows that the atoms are deflected over a range of angles, but why be picky about experimental data that doesn't match a theory? Better to assume experimental error or some type of 'noise'.
Finally, a key problem in Bell tests derives from the fact that Bell's initial analysis (and the Stern-Gerlach experiment) are based on neutral atoms, while all Bell tests are based on photon detection, which, as you point out, are not perfect. I have some ideas about how to translate from atomic phenomena (SG) to photonic phenomena (Bell tests) but your approach is physically reasonable, and may actually be correct for photons. Thanks for a very interesting essay.
I hope you will read my current essay and comment. It challenges another belief based not on math but on physical interpretation.
Best regards,
Edwin Eugene Klingman