If half of a country were impoverished and half were billionaires, I'd say they have a pretty serious income inequality issue. A median would randomly conclude that either the typical person is a billionaire or in poverty. Best case scenario, the median would equal the mean and give a result that reflects reality, but it still doesn't convey the issue at all.

Median and mean both compress n data points into one, which is only useful given some assumptions about the distribution. Quintiles (or similar) would work for that scenario, but I'm sure there are more complicated situations where they fall short.