In my experience, there is a big difference in when the median or the mean is the most useful tool to look at when you want to find the central tendency. The question is what the likelihood of vast outliers is.
A good example is height. The human body can only grow so tall. That means that if you want to find out what an “average” person is in height, then the arithmetic mean is useful. You take a large enough sample, and then you can find with some large degree of certainty that the average height of the population is close the arithmetic mean of your sample. Most likely it will also be close to your median height, as the standard distribution was built on such measurements.
Conversely, some measurements are not bound. An example here is wealth. There is no biological, physical, or chemical reason that a person cannot have all the money in the world. There are people in the world who have so much money that most people in the world could not imagine it. There are then also people who make that first person feel poor. This long tail can lead to some misleading measurements. If you took the wealth of Bill Gates and three random people off the street, the mean wealth of your sample would be around 20 billion dollars. That may sound absurd, but some people have so much money that they can have a heavy weight on the mean of the whole population. That is why economists like to talk about median household income. Currently, if I recall correctly, it is around fifty thousand dollars. The mean household income is much more than that. We use the median here because it gives us a truer picture of the thing being measured.