I understand your point of view entirely, but I think that so far as the present Queen is concerned, she views her position as both a monarch and also as head of the Church of England as indivisible [and also that her position as a monarch is in the nature of an appointment by God]
The problem about attempts to 'democratise' [horrid word] the British Royal Family is that it in effect 'opens a can of worms'. Whilst in the 21st Century is seems unsympathetic and out of line to forbid a Catholic to remain in the line of succession, and it seems out of step to insist on primogeniture, the next logical step is surely to question whether in the 21st Century it is acceptable to have a monarchy at all; the current thinking is that we live in an meritocracy and your birth should not determine your future........... in other words, by trying to make a more 'democratic monarchy' it would appear more logical to abolish it all together...so perhaps better to leave well alone....?
Alex