Is it worth including a binary variable as a random effect within a mixed effect model?

For example, the fixed effect of a binary variable tells the mean effect between 0 and 1 in a regression. If I also include that as a random effect in a mixed effect model, what does this tell me?

For example, things like a rat's sex don't vary from year to year to year. They are fixed across time. But if I included it as a random effect, how would I interpret the correlation and standard deviation.

For example

happiness = sex + money + ( sex | country)

If I am interpreting the output correctly, it is telling me the rate at which the differences between sex are not statistically different. Which seems somewhat obvious and not needed.

In other words, is there anything to be gained by including a fixed effect (non time varying) as a random effect in a model in terms of inference?