How do you handle map projections when the audience is non-specialists?

I keep running into the same problem when I make maps for reports and public-facing work: the projection that suits the analysis often distorts the area badly, and readers notice.

I usually end up choosing something familiar over something technically better, but I'm never sure that's the right call.

How do you decide which projection to use when the map has to work for both specialists and a general audience?