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Median as you're using it doesn't tell us much beyond a very general bit of information.
For example both of these data sets have 5 as a median. But in the second one you would not say 5 is representative of half the country.
[1,2,3,4,5,6,7,8,9]
[1,1,1,1,5,5,6,8,9]
Yes, but the average is HIGHER than the median so it's more like [1, 2, 3, 4, 5, 10, 12, 15, 22]
Well yeah, those weren't meant to be representative. That would look something like,
[1,1,2,2,5,6,7,15,22]
It's more like [2x10, 3x20, 4x20, 5x10, 6x10, 7x5, 8x5, 9x5, 10x5, 12x5, 13, 15, 20, 40, 200]
The whole point of using median is that 0 is fixed, but the upper bound is not, so median is way better than average.
So sure in your example it is not a good measurement, but your example does not represent the real world distribution.
The real word distribution of wealth is actually kind of insane.
And that's forcing people into Quintiles. When you look at the income distribution before the median it becomes very clear it's not a straight slope or at least not in the good way. This was ten years ago. As you can see the median does not represent the mode. Which is what people think of when you say median.
I've never heard of anyone mixing up mode and median, it's always mean and median.
That's because most people don't even know what the mode is. I've met a depressing number of people who think Median is the science term for the colloquial definition of Average. When you tell them the Mode is literally the spot with the most data points they then need the actual statistics definitions of Median and Average explained.
In this case everyone argues over median and average, not even realizing the mode clearly shows an entire section of our country is hurting really bad.