Of course they fear reputational damage! Just how many actual believers does an old, established and very vanilla denomination have left in an educated and multicultural society? Once the CoE was the dominant church in the UK and the Empire. Now it's restricted to the white middle and upper classes, an increasing number of whom are non-religious - including mnay who attend out of tradition or for social reasons. The CoE has no authority, either temporal or spiritual, any more. Its reputation is all it has left.