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It's only in modern times that children usually find their own path in life; traditionally male children would follow in their father's footsteps.
Beau Biden grew up with a father who was a Senator. If his father was a good father, and it seems that he has been, what kid wouldn't want to end up like their dad? Nepotism hinges on an impropriety, rather than just someone who is qualified but also related getting a position.
As for Biden's chief of staff replacing him, to me, that seems like the most democratic short-term solution. Who else but Biden's chief of staff can be expected to share his political opinions and understanding of Delaware's needs? Voters voted for Biden, and in the end they're getting someone who was his right-hand man for years. My guess is voters will end up with someone roughly like who they voted for, which seems about as democratic as it can get considering the circumstances.
Lastly, to imply that connections are somehow inherently bad strikes me as an awfully lonely position to take. Humans have, over history, gathered as tribes, villages, families, cities, and nations because they found themselves stronger together than apart. It's rather antisocial to claim to neither want nor have any connections what so ever.