As one of those Appalachians who did escape extreme poverty, it isn't a problem with an easy solution. I have several family members who are still in one income families, living in trailers, with no desire to make major changes. Some of them are more driven, capable, and intelligent than I am. A couple of them have college degrees, and I don't.
So how did I make it to upper-middle class when they're all still either in poverty or lower-middle? There's no magic formula. I was willing to move to where the jobs are. Appalachian manufacturing has been gutted, so now the area is almost completely dependent on the service industry. Combine that with the easy money of selling farmland to subdivision developers to build homes for retiring Yankees and there's no opportunity outside of the construction or medical fields. Expanded work from home may change that. I'm certainly thinking about returning. The problem with that is that a lot of other people are probably thinking the same thing. Combined with the unending flow of transplant retirees, that makes home ownership even less likely for the natives and continues the destruction of a wonderful culture. No one seems to complain about gentrification when it's happening to the rural people. Funny how that works.