Even in a fully public, NHS-like system, they do. Come to X hospital, no need to go to [big city] for your [surgery] when you can have it here - well, if they don't do this, people will go to the [big city], and if it goes on long enough, the NHS-like system will cut their funding and possibly reassign surgical specialists away due to lack of activity. That leaves the smaller city with less access to healthcare, even if it's free to the user at the point of service.

[EDIT: my karma is now 667. Would someone please remove their upvote just for a bit? A screenshot of "devilbunny (666)" from HN would be a great little digital memento.]