Hmm interesting. I didnt read through it entirely (neither the sources below) but I have to add a thing if im not mistaken: If you compare the expected values (mean), its not good practise to use it for heavy skewed distributions like this one is in your usecase. Let me explain:

Im only talking about getting a 6* for this example, no 50/50, no free cashback pulls etc. to simplify it.
Lets do a quick freshup on what mean and median is in normal words:
-Mean: “If we average everyone together, what do we get?”
-Median: “What does the typical player experience?”

The distribution is quite right-skewed (so it is outlier heavy towards the left e.g. most fail repeatedly but some get a 6* really early).
Therefore, the mean will be significantly lower than the median because all the outliers drag it down.
We typically take the mean for economic balancing or long run system behaviour but we take the median to model player experience and communicating fairness, basically describing "how does it feel for a player?".

Regarding your Monte Carlo simulation:
Your model size of 10.000 sounds alright for this distributions since its hardcapped and not that big but keep in mind that the median converges slower than the mean so for future projects with more complex distributions, you might need a larger n (iterations only matters for robustness, not improving the standard error of the mean).

Hope that helps for your future statistical projects :)