I completely agree that it suggests that we have overlooked something in the underlying state. However, if you take a course in quantum mechanics that includes

I completely agree that it suggests that we have overlooked something in the underlying state. However, if you take a course in quantum mechanics that includes a detailed mathematical discussion of the Bell inequalities (for instance), you will see that any possible deterministic system along the lines that you're thinking of here will give predictions that are measurably different from the predictions of quantum mechanics. These experiments have been done, and the results are incompatible with any deterministic model (regardless of how difficult we assume it is to measure those "hidden variables" without influencing them). I agree that this flies in the face of everything that seems sensible in the world. I loathed quantum mechanics for years for reasons along these lines. Nevertheless, the universe is under no obligation to obey my intuition for what is sensible, nor my sense of elegance and beauty. Much to my disappointment, experiments make it absolutely clear that we do not live in a universe where "common sense" arguments are at all trustworthy on the scale of individual particles. (Also, personally, I would say that the 50/50 result on the x-axis is purely a consequence of symmetry: nothing in the initial state or the measurement apparatus is set up to prefer +x over -x or vice versa, so it would be more or less impossible for it to split any way other than 50/50.)