Now we've got that over with would a Catholic really want to hold the position anyhow? Nothing to do with anti monarchy sentiment at all. I'm a royalist through and through. I do wonder however with the current 'State Church' status of the Church of England (in England- not in Wales or Scotland) whether it would be possible for a Catholic to hold the position when they would be in practice involved in the appointment of senior Anglican clergy and, in theory, the 'defender' of a protestant faith. It's not a position that could be held without compromise. In short, given the current moral condition of the British state, it's not a responsibility that I would wish on a fellow Catholic. Giving the 'royal assent' to an act liberalising the law on questions such as euthanasia or abortion would find a Catholic monarch excommunicated latae sententiae if not ferendae sententiae.
It would seem that the Anglicans have realised this problem and are not in the mood for disestablishment. I hazard a guess, and it's only a guess, that this reflects the private opinion of the Queen who takes her Coronation vows very seriously. As an act of charity to our fellow Catholics I doubt we should be pushing the question. But then it may be the actual agenda but I suspect it's not the real agenda for some leftward leaning Catholics.