It’s subtle but I agree with her on this. Families vs people embeds different connotations. Families necessarily imply interpersonal connection, and for a great many subconsciously conjure children and love.
People does not.
I don’t expect the level of awareness of the distinction from the general population. A politician who spent decades with the highest skilled professional communication crafters on the planet, they know exactly the difference
I would love to live in your world where politicians choose thier words based on logical simplicity rather than rhetorical value.