Reciprocity in Minkowski space is relevant.

In the special relativistic setup in your final sentence, the spaceship occupants could in principle watch our clocks and observe that ours are ticking slowly, with the same redshift.

Observers at a high cosmological redshift cannot watch our clocks at all. They can only look at the even more distant past of what will become our clocks in billions of years, and there's no reason to think that the redshift we ascribe to them is the redshift they ascribe to our precursors.

(Indeed the gravitational redshift from the metric sourced by Earth (assuming "us" are Earthbound) breaks the reciprocity -- if only slightly -- that would be observed in Minkowski space.)