The calculations should be arriving at the same numbers in the end, and implementing a simulation vs. an error propagation tree isn't going to favour the simulation for simple scenarios.
MCM is great for when doing usual error propagation is infeasible. It is sloppy when the error propagation could be done with a one or two analytic formula. Simulation isn't appropriate for problems that are easy analytically. For example, in practice, most of the examples in the article are not suitable problems for breaking out MCM (and although the author probably knows that, maybe not everyone on HN does). The Pi one might be a standard application, but even there Pi in particular is not a great choice for a showcase because Pi has some very nice analytic approximations.
If a modeler has assumed two different models of a situation and are getting vastly different results then that is evidence something is wrong, but it is also evidence that the modeler is out of their depth. It is not appropriate to fit two different models and then claim that the differences are delivering insight. The differences are revealing big gaps in the modelers understanding of key influences, rendering both models highly suspect. They should not be creating models, they should be putting more time into understanding the thing they are modeling.
There are times when a modeler would have two different models, but they should certainly not be surprised that they give different results. Indeed, they should be assuming two different models because they are going to give completely different results. There will probably be other edge cases, but in my experience they are rarer than people just making mistakes about where MCM is appropriate.
or would you rather perturbate your variables and resolve mean and variance. My philosophy drives me towards the computational approach over the analytical mathematical approach.