yes it is just party custom, it would basically just be discrimination otherwise to bar the children of officials from office. But PR China is a party state so the CPC can regulate their officials in ways beyond legal statues. And a princeling can overcome the custom if they are competent, but your work will be held to a higher standard within the party. Xi is technically a princeling, his dad was a high officer who was purged in the CultRev and later rehabilitated. But Xi did good work in state bureaucracy at the lower levels so the party kept promoting him. Xi for his part has kept up the suspicion towards other princelings during the anti-corruption drive.

Western media about princelings is delusional though, talking about them as like some elite aristocracy within the party (projection). The truth is that they are just the generation who was able to get the earliest benefit from the Chinese economy and culture finally modernizing (higher education, new industrial sectors) and as more Chinese people get those opportunities I think princelings will become less relevant all around.