Einmal ist keinmal
You'll Never Walk Alone
...but it seems like Carthage basically just gets to dump all of their generals on the board during the first reinforcement phase on the 2nd turn because the two not already out are considered displaced? Am I correct?
I'm not sure if I'm misunderstanding you, but there are not two Carthaginian generals displaced on the first turn. There are two that start the game subordinate to Hannibal (Gisgo and Mago, I think).
Carthage did not have the political restrictions that the Roman's did when electing new consuls, and instead left their generals in the field. So, you are correct that Carthage begins the game with all their generals in play, and does not have to replace any of them like Rome does.
The Roman player will have at least 2 Consuls and (possibly) a Proconsul in play until the 6th Turn. After that, they will have an additional Proconsul in Scipio Africanus (making it 3 or 4 generals for the Roman player at that point forward).