I know you said public education everywhere might be equally terrible but there are a number of public schools, including boarding schools, for gifted kids. I can't speak to the quality however. Just as an example is this public residential school in Maine. In state residents pay for boarding but not tuition, $8, 450. Non residents pay $30000 for tuition. Moving to Maine might be worth it if you liked that school but there are other states with similar. The focus is STEM in many cases so that is something to consider.
http://www.mssm.org/page.cfm?p=668What about a residential summer course at a Uni if they allow 10 y.o.? They can inspire and introduce kids to a peer group. In the age of mail, video chat etc, peers don't have to live in the same town to be a support group.
