Problems start when you're moving around or doing trigonometry, though; following bearings starts getting weird even at otherwise negligible distances, for example, and runway designators on the opposite thresholds might not be exactly 180 degrees "apart".

But it's not like these effects don't exist everywhere on earth at some scale; they get gradually more pronounced the closer you get to the poles, so I'd be surprised if avionics software would get really confused by them.