Part of them may be the poorer heirs of a onetime upper class (who became not-so-upper-class when losing wealth in Austria-Hungary times or in the Rákosi era), but I mean the culture and heritage that goes back to the really old bourgoise, the urban population preceing the big urbanisation waves.

Can't quite place these... are there really many around?

Yes. Hungary had a major wave of urbanisation from the first half of the 19th century to WWI, especially after the Compromise of 1867. In Budapest resp. its predecessors, the population grew from c. 50,000 at the end of Emperor Joseph II's reign to c. 250,000 at the time of the Compromise and then to 1 million by 1930. Most of the new urban population (former peasants, servants and bankrupted lower noblemen) became working-class, but part of it became middle-class bourgeois (the new bureaucrats, artists, engineers). The assimilated Jewish middle-class dates to this era, too (as does the parallel anti-Semitic 'tradition'). The children and grandchildren of these people would then look down upon and exclude those who entered higher education during the communist era from a peasant or working-class background.

let's say the state sector intelligentsia (and offspring) upon which the left in Hungary almost entirely draws its leaders

The Kádár-era educated state sector was a combination of those who rose socially in the Kádár era, and those from the previous urbanisation wave who could maintain or restore their middle-class status after the Stalinist era. I'm not sure BTW if "intelligentsia" covers all or most of the middle-class: it would appear to me that intelligentsia assumes some participation in public life, dealing with ideas (something even the upper-class can do); whereas you can be middle-class and spend all your professional and free time on issues not touching any of that. If so, I claim that the pre-WII middle-class maintained its domination of intelligentsia through the Kádár era, but the communist-era new middle class had more clue about the economy outside the ivory towers of the intelligentsia. These differences don't matter that much nowadays, though earlier there was some mapping to SzDSz vs. MSzP.

_{*Lunatic*, n.
One whose delusions are out of fashion.}