I'll just mention the game Traitor, by Jonas Kyratzes, which I've recently replayed.
It is an example of a game that has no choice whatsoever, but is still a moral take on an otherwise simple and generic shmup.
So morality can play a role even without choice - but, of course, games are interactive and the possibility of choice is one of the thing that only games can do as a medium, so it's a more important avenue to explore, imo.