Your family may like the far-away schools better when financial aid packages arrive. Private colleges like geographical diversity. They want students from all states. Yet the great majority of students apply to schools within 400 miles.Therefore, if you're further than 400 miles, you have an advantage both for admission and for financial packaging. Apply to a couple "far away" schools, if your child gets into a better school or gets better financial aid to a school closer to home, it's easy to say "thanks, but no thanks" on May 1st.
As for Dartmouth, yes I understand the concerns. :s