That's true, but I suspect the aspects that math can't model are either known to be intractable problems (e.g. modelling turbulence, Navier-Stokes), or otherwis

That's true, but I suspect the aspects that math can't model are either known to be intractable problems (e.g. modelling turbulence, Navier-Stokes), or otherwise have been delegated to other fields like philosophy or the humanities. So I don't think there are physical aspects of reality that we are missing out on because they cannot be modeled using math. However, the social and historical aspect of science is undoubtedly incredibly important, as well as the selection of the scope of science. These aspects of reality might be underrepresented. To be more concrete, perhaps in 50 years we'll have doctors who specialize in "suburban salaryman kidneys," because the scope of study will expand from just the internal organs, to the internal organs, together with the external environment and social conditions. We're already sorta there. A physiologist would conduct research on specific organs and systems, while a general practitioner/family medicine doctor takes your history (i.e. considers the social and historical conditions that lead to illness)