I could well imagine one of the reasons Kiawah gets harder as you progress is the green speed increasing. The poor mapping and nasty breaks are exaggerated by faster greens. Add in playing your approach with a mid iron instead of a wedge and you're looking at much tougher birdie putts.
When I started out I think Kiawah was probably my favourite course, places like BPB and Oakmont were nightmares. Now BPB is my favourite and best scoring course. Oak is still tough but not the monster I used to think it was.