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One personal story NEVER beats a thousand (correctly orchestrated) medical statistics.

Op is probably talking in terms of how to reach a person that has shown already they do not understand science (e.g. they are an antivaxer), not in terms of how to do good science. I suspect they are right in this case, albeit a bit optimistic.

To take this a little further, this framing effect/appeal to emotion has been shown to affect a large segment of the general population, not only those in the anti-science, antivax extreme.

Granted it's not exactly the same thing, but Oxfam uses campaigns which hilight the story of a single child suffering extreme poverty rather than attempting to move people with statistics which express the scale of the problem.

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