Imagine a body with a female hereditary leader, a male chief executive and an all-male board which women aren't allowed to join. Obviously we're talking about a dysfunctional organisation which, you might suppose, is crying out for a challenge under equality law. Now imagine that this body has an automatic right to put its directors in the country's legislature, where they are able to lecture us on controversial matters. Some mistake, surely?

Far from it: what I've just described is the privileged position of the Church of England. And it's why last week's fiasco, in which the Synod failed to pass a proposal to allow women to become bishops, matters to us all. As a non-believer, I'm more than happy to mock men in dresses who represent an organisation which won't allow women to join its senior ranks, but the matter is more serious than that. Even in the unelected House of Lords, the presence of 26 Anglican bishops is an affront to democracy. They occupy places which aren't open to women, as long as the Church continues to have its really quite inexplicable problems with gender.

In effect, the Church is permitted to operate its own male enclave in Parliament, where its representatives vote on matters affecting the female half of the population, including abortion and sex education. This is bad enough, but the Church isn't the only organisation which has failed to adapt to a world in which women expect to be treated equally. Last week the BBC Trust drew up a shortlist of one white man in its search for a new director general, and gave him a publicly funded salary of almost half a million pounds. The BBC plays a more significant role in public life than the Church of England, and this is wrong on so many levels that I barely know where to start.

By comparison, the problem of the Church is easy to solve. Some people defend its leaders by pointing out that they want women to become bishops, but senior clergy have yet to devise a mechanism to see off the hard-core evangelicals who think it has to be led by men. The fact that they're still arguing about it, and about subjects such as homosexual clergy and gay marriage, would be comical if it didn't have a direct impact on how our democracy functions.

The Church's political power is a relic of a period when it had much greater support, but these days its influence is disproportionate in a society where most of us aren't practising Christians.

That's why, after the latest debacle, the argument for disestablishment is stronger than ever. Even some of its staunchest defenders admit that the Church looks like a squabbling sect, incapable of solving its own problems let alone giving moral guidance.

Some organisations find adapting to modernity more painful than others, and a Church stripped of its legal privileges could argue that it's entitled to thrash out its conflicts in private. Time for a retreat, I think, in both senses.

Joan Smith is Political Blonde: www.politicalblonde.com; twitter.com/@polblonde