A hiring-panel member discovers that a close relative has applied. Simply promising to remain objective leaves everyone else dependent on that person's self-assessment. Recusal changes the arrangement instead: the member steps out of the relevant decision so somebody without that conflict can exercise judgment.
For US federal judges, disqualification rules address both impartiality and specified personal interests. An organization's own policy needs a similarly clear account of which relationships require withdrawal, rather than leaving each person to decide whether they feel unbiased.
Withdrawal has to mean something
In an appointment process, recusal might mean not receiving confidential evaluations, leaving the discussion, and not voting. Merely abstaining at the end would accomplish little if the conflicted member had already steered the conversation or privately pressured colleagues.
The group also needs enough unconflicted people to make the decision. In a small community, relationships are common, so an outside reviewer or alternate panel may sometimes be necessary. Otherwise, a recusal policy can leave nobody authorized to act.
Recusal protects a decision's legitimacy by changing who participates, whereas disclosure only makes the conflict visible. It is a targeted response to a particular matter, not a declaration that the person is generally dishonest or unfit to serve.