The personal faith of PMs

I have seen it suggested that because Rishi Sunak is a Hindu, it would be wrong for him to have any role in the appointment of bishops in the Church of England. This is a non-sequitur. So long as the C of E remains the church by law established, its main appointments must, in formal terms, be made by the Supreme Governor of that church, the monarch, ‘on advice’. That advice, though informed by the views of the hierarchy, must be tendered by the Prime Minister. Gordon Brown, when prime minister, tried to shuffle off these responsibilities, perhaps feeling rather Presbyterian about them; but this does not work. If the Church of England has the privileges of Establishment, it must permit an exterior eye on whom it appoints. And there is no evidence that Anglican PMs make better ecclesiastical appointments than non-Anglicans. Indeed, I suspect the reverse may be the case, since a strong Anglican will tend to have his/her preferred church faction.