That isn't possible because the social structure she advocates breaks the moment wealth/empires are passed on to the next generation.
I disagree.
No, her point was that self made men need to be unfettered by the restrictions of their lessers in order to create a glorious world and enjoy their justly won wealth.
You are glossing over something very important: that there are necessary, just laws (against murder and theft, for example) that all people equally are subject to, and that there are unnecessary, unjust laws (one group lobbying and getting the ability to steal another group's property or unfairly impeding a competitor through deals involving cronyism, for example). Rand is in favor of the former and against the latter. Your statement is only true with respect to restrictions in the latter category and incorrect if you mean it to apply to both categories.
She fails miserably at it too because she doesn't account for human nature. . . .
Her books revolve around human nature, namely that if allowed, people end up using coercion to steal things rather than through free transaction. This is the same thing you are complaining about, yet she addresses it directly, and it is the central theme of her works and philosophy. Your feeling is that the first generation involved in setting up such a system will be OK, but their children will relapse into corrupt kleptocracy again. Well, there are no systems that escape that the potential problem of degeneration. To avoid degeneration every system -- hers, the ones the Romans used, the ones the Greeks used, the one America was founded on, communism, etc. -- has to be upheld by people. This is a given from the start.
Rand was fundamentally opposed to democracy.
True, as long as you understand that the definition of "democracy" in this case is what the founders of the US were also opposed to (which is why we have a republic, electoral college, House, Senate, Excutive, Legislative branch, etc.).
An Ayn Rand quote:
"'Democratic' in its original meaning [refers to] unlimited majority rule . . . a social system in which one’s work, one’s property, one’s mind, and one’s life are at the mercy of any gang that may muster the vote of a majority at any moment for any purpose."