Many of the other comments deal with thought experiments rather than looking at the reality of how “many worlds” is USED. From my point of view as a non-physicist it seems to primarily be used as psuedo-science “woo”—a revival of mystery and awe under the cloak of scientific authority. A kind of paradoxical mysticism for non-religious people, or fans of “science-ism”.
An agent might act differently from MISUNDERSTANDING many worlds theory. Or by paying more attention to it. Psychological “priming” is real ansd powerful.
The answer by TAG below is case in point. For someone committed to a belief in determinism or fatalism, having a manyworlds theory in mind may buttress that belief.