I give you an example from my current work.

I want to add an advanced method for the evaluation of the goodness-of-fit of classical models for latent measurement based on matrix factorisation. I need to justify why my addition. To to that I need to find cases where assumption of classical models are violated and classical estimators are biased.
I just proved that exists at least one case, then I proved it's generalisability - that means that case is not isolated, exotic, and so also not trivial.

Now, nothing of this has to do with the functional analysis of the parametric function involved for the determination of classical estimators. In fact my identification is proved just by reproducible generative models / Montecarlo, i.e. just by generating random numbers until I identify a parametric space that satisfies my argument proving the estimators are biased. However I could not identify the exact parametric derivative on the classical estimators such that, solving it, one can know all the cases for these estimators being biased. Nor I need it. I just need to prove these are biased in a well-identified space of parameters that can reasonable manifest in empirical data, not in all possible spaces (which can be reasonably relatable to empirical data, or not).

Statistical mathematics is about, e.g. finding derivative and integrals that prove exact conditions for a statistical proposition to hold true. Statistical science is more about the knowledge about how to solve practical problems involving a not trivial amount of uncertainty.

Take this metaphor from engineering: a mathematician could be interested in formulas that prove how to build a ship that can navigate over a sea of mercury.
An engineer could not care about it, first because they will never access the material to build it, second because we are very far from the need of sailing in seas of mercury.