Design a sampling process to select an element with probability proportional to its appear probability in a simulation
This can be done efficiently if the size of the samples $S$ is not too large.  Let $m$ denote the maximum possible size of $S$. Then the following procedure outputs exactly the correct distribution:


Draw a sample $S$ (using the black box $A$).  With probability $|S|/m$, keep $S$ and go to step 2.  Otherwise, go back to step 1 and draw a new sample.
Choose an $x$ uniformly at random from $S$.  Output $x$.


You can verify that this yields the correct distribution.  It requires at most $O(m)$ samples (possibly less, depending on the distribution of the size of $S$).



On the other hand, if the samples can be very large, then approximating the distribution accurately may require many samples.  Consider the following situation. Randomly choose a subset $X_0$ of $X$ of size $|X|/2$, and imagine a black box $A$ that works as follows: with probability $1/s$, it outputs the set $X_0$; otherwise, it outputs a single element chosen uniformly at random from $X$ (i.e., a random singleton set).  Then if you draw $s$ samples, you have about a $1-1/e \approx 0.37$ probability of hitting the large set $X_0$.  Moreover, if you don't hit the large set, the probability distribution of your procedure will be off by a total variation distance of $1/s$ (the probability for each element will be off by an additive error of $1/(s|X|)$).  So, if you want a distribution that is close to correct in total variation distance, you'll need many samples. If you want an additive error of $\epsilon$ with probability at least $1-\delta$, you might need something like $O(1/\epsilon)$ samples (I'm not sure about the dependence on $\delta$).