The fact that they're not upper-upper doesn't change the fact that a public school education is a fast track to Oxbridge and to insider status.
But even with the fast track, the upper middle classes are motivated by anxiety and envy about not being upper upper enough. The original article almost highlights this, but doesn't quite hammer it home.
The class system is about aspiring to become the kind of person who has ancestral money and some huge houses. (Not just one huge house. That's the entry level.)
But the people with the ancestral money and huge houses are neither special nor interesting. They're groomed for confident self-interest and social polish, but they don't actually do much except cream money off the markets, order inferiors around, and breed.
Their self interest sets the tone for those lower down the tree who aspire to be like them. That's why they're so toxic. They're hostile to any kind of collectivity which erodes their "natural" privilege.
And they'd rather own a country which is falling apart than lose some status in a country with more social mobility which might challenge their self-perceived high status.