The Solution!
Those who were thinking things about subdivision were absolutely correct. The answer came from user julperado in this post, which I will quote in its entirety (with one parenthetical insert):
If you plan on having clothes that fit the body, then you can model the clothes normally and use the Surface Deform modifier to follow the body deformation, without having to bind the cloth model to the rig or run a sim. The process is this:
Make a copy of the body model and rename it to something like “deform_mesh” or similar, just so you know that’s just a reference for the modifier. Make this object not visible in render.
Remove any subdivision or multires modifier from this mesh, but not the armature modifier [or, in my case, the mesh deform modifier]! The idea here is to remove anything that modifies the polygon count. (This step is important because the Surface Deform modifier uses the original polycount of the target mesh, so if you bind it to a model that has subdivisions, it will not render properly because the polycount in render time will be different to the polycount used to bind your objects)
Now select your cloth model and add the Surface Deform modifier, on the Target option select the duplicate of the body mesh, Interpolation Fallof and Strenght can be left unchanged, those defaults work fine most of the time.
Now click on Bind, and your clothes will deform exactly as the body without doing any extra rigging or simulation.
Keep in mind that this modifier will work better in lowish polycounts, so if you’re going to use Solidify, Subdivision Surface, or Multires, add it AFTER the Surface Deform modifier.
Hope this helps
EDIT:
This process can also be used along with a cloth sim; on a dress or skirt for example, you could bind the mesh only on the waist area using a vertex group, and then run the cloth sim for the rest of the mesh, so it will follow the body deformation and the cloth sim will react to that.
I never tried entirely disabling all subsurface on the body mesh in final render, as the final render should have that enabled.
So with the body duplicated, and that duplicate body removed from render along with all vertex-count-modifying deformers removed, then jeans, socks, etc all bound to it works properly! (It is still bound to both the cage deform and the armature, just as the original, so it deforms the same as the original.)
It is nice to understand what the problem was and have a solution! 
I was so slow…
No image either, it looks the same as with the deform cages (as it should). Not sure if I am going to update everything as I currently have a working solution with the Mesh Deform cages, but it is a very easy change should it become necessary.