While some people crave being in charge of others, many shy away from it. The pressure of being responsible for someone else's well-being, success, happiness, etc. can be so great that a character will actively avoid being put in this position.

A fear in this area could realistically develop out of a wounding event—whether the character failed when they were put in charge of someone or they were the one who was let down by the person responsible for them. Their reluctance can also stem from other fears, such as a fear of failure or disappointing others.

Not having children
The character avoiding situations where they're put in charge of others
Difficulty building deep relationships with others
Taking jobs that allow the character to work alone
Shifting responsibility for others onto someone else
Maintaining a persona as a poor role model
Gravitating to friend circles, work groups, and organizations where someone else is an A-type personality who likes to take change
Acting irresponsibly (to keep others from considering the character when someone needs care)
Avoiding leadership roles
Being neutral or apathetic about social and political issues (because if the character expresses concern or notices injustice, they'll feel compelled to take action for those being mistreated)
Claiming that social problems aren't real or are someone else's problem
Claiming that people needing help are responsible for their own misfortune (so the character can avoid taking responsibility)
Selfishness (real or perceived), because the character's priority always seems to be themselves, their own needs and desires, etc.