The Animal Within

At the age of sixteen, everyone presents as Alpha, Beta or Omega. Percy was always expected to be a leader - as a child of Poseidon he was unlikely to be anything else - so it wasn't a surprise for him when he turned out as an alpha. He doesn't think much about what's expected of him, until he's almost buried alive with an omega. The gods can hardly blame him for what happens after. Of course, his intended omega will not make it easy for him - there's also the question of his very territorial brothers.
Then again, asking Kronos to roll over and play nice would be like asking a dolphin to walk on land - and Percy wouldn't have him any other way.