A better way to view this is: replace the word "stable" with "stable under pressure."
That just seems to me to diddling the rules; a neutron star isn't under external pressure; it is in free space. The important point is that a nucleus has to become quite large before the gravitational force begins to play a role in its stability.
Another thing is: the usual nuclei generally are in the ground state. Excited states that last long, are rare for nuclei. But for a neutron star, it presumably never reaches ground, there are too many excited states it gets trapped in for long periods.
That and related things do seem to be major differences between `small' and `large' nuclei. A neutron start presumably changes both atomic number and atomic weight quite freely, so you could reasonably argue with its being stable. Whit