Institutions need irrationality. It is possible to object, on rational grounds, to just about anything. But to point out that, say, the Church of England [or the House of Lords, or the monarchy, or any other example you care to throw in here] is pretty much indefensible as an institution, is to miss out on the curious truth that [insert your example or mine] does serve a purpose. Invariably - here's the curious part - it's not the purpose that we associate with the institution. But it is a purpose that needs to be served.
I remember the survey put out a year or two back by Professor Richard Dawkins on religious belief. Very roughly, Dawkins' contention was that a lot of people claiming to be Christian didn't seem to know very much about Christianity - not knowing, for example, which was the first book of the New Testament. This, suggested Dawkins, invalidated their claim to be Christian (I'm telling this from a combination of memory and YouTube). Challenged by Giles Fraser in a radio interview to give the full title of Charles Darwin's Origin of Species, Dawkins muttered, "Oh, God!", and couldn't. Put <Dawkins Fraser> into a search engine and find what I just found on YouTube.
Dawkins subsequently wrote at length in the New Statesman about why this didn't matter, and I sympathise. But what I took away from that episode was Dawkins' survey's finding that a large proportion of people self-identifying as Christian said that their faith made them want to be better people. Not bad: the Church of England/Christianity has a positive effect on society. It's useful in that it makes people want to be better people. Although a certain well-known Deity rarely gets a mention in its deliberations.
If we can forget their stated purpose, institutions can be useful. The reality of monarchy is minibuses, but pageantry boosts tourism. The higher reaches of the Church of England may echo with debate about whether women can dress up as bishops, but see above - and the local clergy are pretty good at community events. In fact, you know, it strikes me that some of the most dangerous people on earth are the true believers - I'm talking mostly, but not only, about religion. Maybe it's a good thing that nobody in the C of E mentions - you know who.
So maybe we should just nod, and smile, and agree that if they want to look for the Word of God in the small print of who can wear what, well, that is indeed where they'll be looking for it. Pointless activity, but without it, would there be anybody to splash water on the heads of our babies? Or listen to the elderly in their last days? It's all indivisible, and irrational, and so what?