I am probably a fundamentalist atheist, so let me put that out there first.
I grew up in a village where most activities were either directly or indirectly tied to the church. So the hall belonged to the church, the school of CofE, and all the festivals were well attended. I go back now, thirty years later, and there is one vicar for five village churches, because attendance has disappeared. The school is still there, but no longer CofE, now an MAT non-denominational.
Whilst the RF have a centuries long association with the Church, most people in my life are atheist, agnostic or humanist. The Monarch has to be Defender of the Faith. I don't know if a closet atheist can be a Monarch, or if they have to abdicate...but some previous holders of the post were hardly moral or exhibited pious behaviour.
I believe in the disestablished CofE, and for the Bishops to leave the House of Lords, to stop them influencing laws-making and trying to cling on to power within the Establishment.
I firmly believe that the CofE can exist happily separate to The State, because there are still parts of the country where they are still the drivers of culture and history. A lot of the King's duties are linked with anachronistic historic traditions, but there's enough support for these to keep going.
As for me, an ordinary citizen, the CofE only touches my life when I visit their buildings as a tourist, as I see them as historical landmarks rather than religious ones. I accept the CofE as part of our past, but not our future. We are so multicultural now, the CofE cannot claim to be the leaders in religious matters anymore. And if the bishops must stay in the Lords, then so must a Rabbi, an Imam, a Guru, ...