Or think of it another way. It's not about making the same amount for each nurse regardless of location, it's about nurses enjoying roughly the same quality of life regardless of location.
At this point, most nurses make enough to live middle-class lives. We're not rich, but we can generally afford decent housing, keep food on the table, have reliable transportation, pay for daycare/children's activities, and take modest vacations now and again.
So imagine, we got rid of the outliers (high pay in HCOL areas and low pay in LCOL areas) and paid every nurse the same median wage.
Well, the nurses living in MCOL areas would go on as before. The nurses in LCOL areas would get a nice little windfall. They could save more, go on more vacations, buy bigger houses, etc. The nurses in HCOL areas would be screwed. They'd be relegated to tiny apartments (and or ridiculously long, draining commutes), rarely get a chance to eat out, never vacation, and not even be able to afford daycare.
So if they're all working just as hard, why should nurses in LCOL areas live the high life while nurses in HCOL areas barely scrape by?
You know what happens in those HCOL areas, if conditions continue like that? Nurses quit. Hospitals can't attract or retain staff. Patient care is endangered. So, to attract the staffing they need, hospitals have to in crease pay so their nurses earn enough to live a moderately middle class life. It's really a simple case of supply and demand.