Entrenched aristocracies happen when you have old money that looks down on the "nouveau riche". Every country has an old money class that considers itself more elite than the nouveau riche, of course, but they aren't really an entrenched aristocracy unless the rest of society agrees with them.
That's not the cause of aristocracy, that's the effect.
The historical exception is the U.S. from 1945 to 1973, as well as perhaps a few other examples (maybe Scandinavia today, or the U.S. in the early/mid 1800s?). In that time period - really, no more than a generation - we saw a big leveling of income distribution, where people born to modest families suddenly had the creature comforts that a generation before had been reserved for the aristocracy. I think most people would agree that this is a generally desirable state for a society, so it might be worth asking what was different about those time periods that made them the historical exception?