I suspect this is why there are (and will probably never be) cubes of
orders greater than 5. I believe (though have not proved) that the 5 cube
contains all the complexity that is possible. Adding more cubies would only
increase the amount of time needed to solve.
On the other hand, a 5X (or any cube of odd order) will still have the
constraints imposed by a fixed center. As a single example, the 4X here in
my office is completely "solved" except that two opposite corners are
swapped. That's not something that can happen on a cube of odd order (at
least I don't think so, but I would love to be proved wrong ;).
Wow! I could have sworn I have gotten to this position before, but you are
very definitely correct. The state with two diagonal corners swapped is in
the orbit with edge cubies exchanged.