responsibility, duty, term, entrust to, appoint
任 centers on being given a task or role and carrying it out: taking responsibility, fulfilling a duty, or being entrusted with a position. The sense of 'term' follows naturally from an appointed period of responsibility.
duty; responsibility (incl. supervision of staff)
to leave (a matter, decision, etc. to someone); to entrust (to someone); to entrust (someone) with
duty; function; office; mission; task; role
optional; voluntary; arbitrary; random; discretionary; facultative; spontaneous; any
assumption (of office); taking up (a post); inauguration; installation
resignation (from a position)
person in charge; senior staff; manager; chief; head
being in charge of (esp. a class or subject); taking charge of
appointment; nomination; ordination; commission; designation
dismissal (from a post); discharge; removal
to entrust (to someone); to leave (in someone's hands)
term of office
obligation; duty; charge; responsibility
moving to a different location to start a new job; (proceeding to) new appointment
entrusting; charge; delegation; authorization
successor
full-time service
retirement; resignation; stepping down
serving concurrently as; holding the additional post of
nomination (of a person to a position); selection; election; assignment; appointment
standing; regular; permanent
trust; confidence; credence
newly appointed; inaugural
successive jobs; consecutive jobs
任 combines 亻 (person) with 壬, which likely contributes the sound and may carry a sense of bearing or carrying. The character expresses a person bearing a responsibility or being entrusted with a task.
A person 亻 carrying a heavy load 壬 on their shoulders: that person has been entrusted with a responsibility and must fulfill their duty.
For ニン, imagine a ninja entrusted with a secret mission: ninja -> ニン, and the ninja takes full responsibility for the task.