Yes you "could simply draw pairs from $X$ and $Y$ an arbitrary number of times, and report the proportion of instances in which $x > y$." What you wish to estimate is not a random variable; hence, it is constructed from a point estimator of a parameter of a random distribution. In this case the random distribution may either be the difference statistic $D = X - Y$ or the ratio statistic $R = \frac{X}{Y}$ and the parameters would be $P(D>0)$ and $P(R>1)$, respectively. Clearly both of these are equivalent to $P(X>Y)$. Looking at the posterior distribution of $D$ and $R$ will provide additional information into disparities of the posterior densities of $X$ and $Y$, but perhaps a simple overlay of the posterior densities would suffice.

Yes you "could simply draw pairs from $X$ and $Y$ an arbitrary number of times, and report the proportion of instances in which $x > y$." What you wish to estimate is not a random variable; hence, it is constructed from a point estimator of a parameter of a random distribution. In this case the random distribution may either be the difference statistic $D = X - Y$ or the ratio statistic $R = \frac{X}{Y}$ and the parameters would be $P(D>0)$ and $P(R>1)$, respectively. Clearly both of these are equivalent to $P(X>Y)$. Looking at the posterior distribution of $D$ and $R$ will provide additional information into disparities of the posterior densities of $X$ and $Y$, but perhaps a simple overlay of the posterior densities would suffice.

It is important to stress that to draw a sample from the posterior density of $D$ or $R$, one merely draws a random pair from the posterior density of $(X,Y)$ and computes $D=X-Y$ or $R=X/Y$.