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Rather, Von Neumann often discussed the measurement of the spin component of a spin-1I2 particle in various directions. Clearly, the possibilities for the two possible outcomes of a single such measurement can be easily accounted for by hidden variables [ ... ] However, Von Neumann felt that this is not the case for many consecutive measurements of the spin component in various different directions. [... ] A great number of consecutive measurements will select particles the hidden variables of which are all so closely alike that the spin component has, with a high probability, a definite sign in all directions.
9, p. 699] Bohr is careful to discuss position (and momentum) in these terms, not speaking of the position (or momentum) of a system, but its position relative to some other system that serves to define the frame of reference in terms of which position and momentum are defined. Note that Bohr is not insisting on an operationalist definition of position, here. He is not claiming that position is defined by the operations used to measure it. He is simply (and much more plausibly) insisting that a well-defined frame of reference is crucially a part of the notion of position.
The second point does become important when we consider instead the case where we measure PI. Bohr claims that such a measurement removes the possibility of determining the location of the diaphragm relative to the support. Bohr's argument is-to put it kindly-' compressed' , but I suggest that he is arguing that because the arrangement requires the diaphragm to move freely with respect to the support (lest we be unable to determine PI + P2), the only way to determine the location of the diaphragm relative to the support would be to measure the position of one of the particles, relative to the support.