I've always hated the look of the Morgan headrests, I think they're ugly and they spoil the lines of the car. I've always taken mine off and put the little plastic MG bungs in the holes.
I accept that a properly designed headrest in a modern car is a life saver, preventing whiplash. In conjunction with forward facing and curtain airbags, seat belt pretensioners and crumple zones designed to absorb the force of any impact, modern seating is part of the safety system.
In a trad Morgan there's so many other things likely to cause you serious injury in a collision that worrying about headrests seems like wasted energy to me.