You can't check for causation unless you manipulate a subsample of the companies randomly.

Article: "However, the authors cite studies of changes within individual firms and organisations which seem to show that improvements in employee morale precede gains in productivity, rather than the other way round."

A theory could be that employers see the improvement in productivity before they are measureble by the methods employed and therefore are happier. E.g. bad processes are replaced which gives happier employees before the gains are evident.

Right. Sometimes you can eliminate obvious correlates, but it's quite difficult to eliminate all of them.