Normality denied

See also: reluctant hero

This is extremely common in eastern fiction (especially Japanese), where the protagonist gains or has some Terrible Responsibility due to their powers, uniqueness, or whatever.

Likely a not-so-subtle attempt to make (young) people want to live normal lives, or a bad attempt at making the character seem deeper.

There's at least two possible ways to present this. Either unwanted power, or unwanted responsibility, the former usually implies the latter but latter can exist on its own (e.g. a highborn protagonist rebelling against the responsibilities of nobility while they want to be a "simple commoner"). Can have various thematic elements, including: tragedy of having Powers (makes you a magnet for disaster), responsibility (and the damage avoiding it does), and so forth. A common selfish theme is where the powers come in way of the character's earlier goals/dreams. May involve growth into acceptance of the power/responsibility, or if tragedy, the Great Disaster caused by avoiding it, or finally being freed of the power/responsibility (redemption of sorts, finally being able to make their original goals/dreams come true).