In my opinion mechanical designs for the 7 and above will have to be
fundamentally different from those for the 6 and below, because that's
the point at which the "buried" corner of a corner cubie extends past
the surface of the face during a face turn and thus it's not possible
to build the thing as rigid pieces connected to a central mechanism, at
least not without cutting away part of some face-center cubies.

One solution to this dilemma is to let some of the "cubies" become
"brickies" (i.e., rectangular bricks instead of cubes). In this approach,
there's no limit in principle on N to how large an NxNxN puzzle you could
build with the standard mechanism. There is, of course, the lower limit
you just described to how small the corner cubies could become.