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Some of these might fall into:
With Great Power Comes Great Responsibility - Someone who's got the power to change things for the better feels a push to use that power to change things for the better. It's rarely something that just happens immediately. They may be a little selfish or passive at first, then they have an epiphany at some (early) point and vow to only use their powers for active good.
Anthropic Principle - If someone has super powers but decides to only use them for convenience, The End, then that's not a very interesting story. But getting into danger, that's interesting.
Messiah Creep is the first thing that came to my mind too, but I think that's a bit more specific. Might be a missing supertrope.
Thanks, guys, Messiah Creep has most of the examples I was thinking of. I think Anthropic Principle is a more complicated trope, though, since it's the idea that, for a story to happen, characters need to be caught by circunstance even if it's not "realistic" because the story's being told about them. But having a character assume the role of hero because they keep getting into trouble feels more like an Averted Trope twist on the idea.
Some of this is just Character Development, too- the one who was Not in This for Your Revolution coming to care about higher goals than their own self-interest, that sort of thing

Is there a trope for characters drifting closer to the idea of generic crimefighters as a series goes on? It's a bit general of an idea, since it comes in many different ways, such as: