Easy way to refine a 3D-model for 3D printing by removing internal geometry
About 10 minutes after writing the question, it suddenly dawned on me, that I was not using Meshmixer to its full potential, and especially not a simple property of Cura:


Cura can discard all parts of a model, that are intersecting a closed volume and thus are inside another volume - fixing intersecting shells this way. This option is called "Union Overlapping Volumes"
Meshmixer can, using the meshmix tool, add simple and somewhat complex geometries to specific places in given size and orientation.


Now, these two parts can be used to get rid or complex internal geometry by intentionally setting it up an intersecting shell. Usually intersecting shells are a no-no for good 3D design, but by having something - for example a cylinder - intersect just the internal geometry and enclosing it, these parts vanish in the slicing, if the slicer is set to fix intersecting shells by ignoring internal geometry. Cura does so with a simple setting, one that seems to be on by default in the 3.3.1 distribution. 

So the odd solution to how to reduce a model's internal geometry without impacting the outer look can be this at times: 


Manually add geometry, that (fully) encloses the internal geometry and turns it into an intersecting shell, allowing (some) slicer softwares to ignore this part.


To illustrate, almost the same area of the object before and after the added cylinder:

   

As one can see, the structure became much simpler, as the cylinder cuts away all the internal structure (the 'spokes' and 'axle' one might want to identify from the left picture, but that would be a misidentification) is gone. Much less internal geometry is retained and instead it is now filled with a cylindrical space of nice and fast(er) to print infill, here "Quarter Cubic".

While this is just a partly automated solution - demanding the manual addition of the intentional intersecting shell - I am yet to be taught about a fully automated way.