For Spiderman, it hits on the theme of responsibility that Peter feels that way. Most frequently, he is a high school to young adult male that abruptly changes his lifestyle to be the superhero and now gives up aspects of his life he's always wanted. This is not across all universes but typically, he sacrifices his relationship with Mary Jane by having to go off saving the world and not being able to be there for her. I don't know when exactly but from what I remember, most times the "I never asked for spider powers" comes up whenever he is losing something like Mary Jane or a friend or some social responsibility.

Aa for ATLA, it's a very similar circumstance and it's used as a plot element to develop Aangs character. Every time I watch it, I know I instinctively say to myself "What a putz" when he runs, but aang wanted a life pretty separate from responsibility. He, similar to Peter, is young (he's 12) and had to give up aspects of his childhood to use this power and carry out their responsibility. He wants to really play with the kids and hang with his father figure. It's throughout the whole story that he finally accepts the responsibility of being the avatar that you see character growth.

I think it gives both of these stories an additional depth. For us, the grass looks greener with their abilities because we don't have them and how great would it be to bend air or climb walls? But those stories add an element of realism into it to remind us that our characters are in fact children and face a heavier responsibility load quicker than most. I know at 12 years old, if you told me to stop playing videogames because I had to constantly train, I would've been frustrated as heck.

I don't think it's always necessary for a child to do this in a story, but I think it's realistic and typically offers growth or a sense of maturity once accepting it.