Either way, mean (as in per capita GDP) is probably not the best measure of central tendency for showing how the typical person lives.
Also, median also has issues dealing with things like the US's vast undocumented workforce. So, you really want to slice things up by income percentile, family type, PPP and non etc.
The dataset plugged in and the mathematical operation performed on it are two distinct concepts and shouldn't be confused. It's the same reason the IRS lets undocumented citizens and documented citizens performing unsavory work (including drug dealing and prostitution) to file taxes :)
For an extreme version southern slaves made ~0 income but because they made up less than 50% of the population they disappeared from that statistic.
Their presence in the tally means that the median skews their way. For each undocumented person earning under minimum wage, the median shifts one person over to the left.
I agree that there should be a better way of displaying that info (at least quartiles) but the median is a surprisingly effective measure.