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I am not educated about particle physics or whatnot, but isnt the uncertainty principle momentum and position cannot be both estimated to arbitrary accuracy? so one could estimate only one of them to arbitrary accuracy and then 1000 particles increases the confidence of that position estimate?

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Position and momentum are not separate things that you can measure separately and recombine. They're deeply interrelated.

You would improve your counting statistics somewhat with 1000 particles*, but basically you'd just resolve the Gaussian distribution more clearly. The uncertainty of the measurement comes from the width of the Gaussian, and it's not like measuring more points is going to somehow make that distribution skinnier.

*(a hilariously, impossibly small number in quantum world, where photons come in doses of 10^20/second, or electrons come as 10^18/second)

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