How does it select leaders? By finding the most talented, or by finding those who are controllable?
Seems like the institutions are dual use.
David Cameron - Eton, Philosophy, Politics and Economics at Oxford, went straight into politics.
Theresa May studied Geography at Oxford, worked at Bank of England and a few other institutions for a while.
Boris Johnson - Eton, study of the Classics, ancient literature and classical philosophy. Worked as a journalist, was fired for lying, then became a politician.
How many politicians are hired or promoted by prime ministers not because of quality or background but social debts between families? Nobody really tracks those relationships.
Eton isn't regarded as a particularly good school. It is relatively expensive, and preferred by people outside the UK. Their academic results aren't that strong.
And social connections are an aspect of almost every political party. Blair hired almost everyone in Derry Irvine's chambers. Most of the SNP are not only friends, they are actually related to each other. The Tories are possibly the least incestuous because Cameron made a concerted effort to widen the reach of the party in the mid-2000s (this is why the current Cabinet is so diverse).
The reason why there is a concentration of Oxbridge is because, when most of these people went to uni, they were far and away the best. In Scotland, you see the exact same thing with Edinburgh (and strangely, no-one talks about the fact that almost of Sturgeon's clique went to Glasgow...which is, generally, an inferior university). And what you are actually seeing now is that more and more politicians are going to other universities reflecting the fact that non-Oxbridge unis caught up in the 90/00s (and recently, they have dropped off far more...Imperial is ahead in sciences, Oxford PPE is still very good though).
Looking at Instagram there are undeniably many physically fit Americans. In reality their physical fitness is notable because it is relatively rare.
Elite universities are sustainable in an industrial society where scarcity of resources means there are only so many high positions.
In a post-industrial society knowledge is the resource. Elite universities then instead mean you have a lack of resources.
Err, no.
Don't get me wrong, the Oxbridge system accepts, fosters and matriculates some very bright, very smart people.
But that is not at all what the core of the "Elite University" system is about, it's about connections and maintaining a single degree of freedom between the families that collectively hold > 90% of the wealth, resources, and influence within UK society.
Pretty much any male politician whose official actions could be described as inhuman or sociopathic has been through boarding school trauma.
In a post-industrial society, or beyond that, you aren't tied to physical resource in the same way. Instead you are limited by the amount of people with relevant knowledge. The existence of elite universities in such a society is really more saying that you can't produce enough people and opportunities.
Arriving at a village finding there is only one person with a car you don't say "wow, this guy is successful" you say "ouch, this is a poor village" even if that is the most successful person in the village. Because these days we expect cars and if everyone had cars he wouldn't stand out as successful.
The unfortunate thing is that we have been governed by a party which takes that unhealthy attitude towards senior party members, they have always been more about who you know than what you know. You don't see the same proportion of Oxbridge grads in the current opposition.
Slightly anecdotal, my first job after graduating (from former polytechnic Loughborough) was in Cambridge for a company formed by a couple of Cambridge grads. There was a higher proportion of non Oxbridge grads at the company, I think we tended to have more practical experience.