In the UK there is definitely a perception that the Civil Service is the preserve of privileged Oxbridge graduates, possibly even only those who came from public schools (Eton, Winchester, etc). It is debatable how true this perception is in the modern day, but certainly it was felt strongly enough in the past to be satirised in television (most prominently in, say, Yes Minister and its sequel).
There is the definite feeling that the Civil Service is or was a sort of old boys' club, where those with connections would head to after finishing their classics degrees at Oxbridge. Attending Oxford in the 2010s, the idea of the Civil Service being a natural next destination was certainly still prevalent.
I contrast this to the equivalent in the US: as far as I can tell there is no association between the US CS and, say, the Ivy League. The perception in the US seems to be the polar opposite, with almost a sense of disdain by Ivy League graduates of the idea of working as a bureaucrat as opposed to entering the corporate world.
What were the historic factors that led to this perception in the UK, and to what degree was it ever true?