The mythic status of the NHS results from it being both the modern institutional embodiment of the Sherwood Forest redistributive project and possessing a very strong connection to the last major (pre-Brexit) enactment of the dragon slaying: the defeat of Nazi Germany.
The dragon is of course a metaphor for any tyrannical foreign otherness that threatens the community. Since the end of WWII the European Union has gradually come to assume elements of this role in the English psyche.
This explains why liberal-remainer ideologues have a problem getting any message across beyond the practical difficulties posed by a dead dragon in this instance.
They have their own myths, not without serious contradictions, but these don’t make such an easy fit with the two above-mentioned national tales.