When to test For Equality of Medians, and when Stochastic Equality?

In statistics, we are very often interested in investigating whether some score (let’s say life satisfaction) tends to be bigger in one population (let’s say rich people) compared to another population (let’s say poor people). Most often, this research question is formalized by testing the null hypothesis of equal means. However, for a comparison of means to be sensible, the variable of interest must be at least on an interval scale. So, for ordinal data, other formalizations should be used.

Two formalizations are generally used for ordinal data. Equality of medians and stochastic equality, which is defined as $P(X<Y)=P(X>Y)$, where $X,Y$ are the random variables representing the two populations. Many papers argue that stochastic equality is the better formalization. The core argument is that the medians can be equal even if the scores in one population are clearly bigger. Consider as an example the following mixture distributions for $X$ and $Y$. With probability .5 $X$ is just $1$, and with probability $.5$ it is sampled from $text{Uniform}[0,1)$. Similarly, with probability .5 $Y$ is just $1$, and with probability $.5$ it is sampled from $text{Uniform}(1,2]$. Thus, $text{Median}(X)=1=text{Median}(Y)$, while no realization of $Y$ is smaller than any realization of $X$ and $50%$ of the realizations of $Y$ are bigger than all realizations of $X$.

Thus, should we stop testing for equality of medians, or are there research questions for which the equality of medians is the appropriate question? If yes, what are those?