The expansion of the universe is so tiny that it is insignificant compared with local gravity. It would be like trying to factor in the gravitational pull of Pluto on an object as it falls from your desk to the floor; technically there would be a contribution, but it would have no meaningful effect.
And this doesn’t just apply to other stars within our galaxy; it applies to galactic clusters as well.
Universal expansion is only a meaningful thing on the scales between galaxy clusters. It happens on scales smaller than that but not to any degree that is worth caring about or is measurable, even over non-trivial timescales (at least for now).
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