The OP is right about the recruiting efforts of the private universities. It's not just the rural schools they skip. My son went to a high school near San Diego that straddles the border between an middle class area built in the 50s and a low income area. The student body is about 40% suburban white, 25% Hispanic, 10% recently moved from Iraq (Chaldean), and the rest African American, Native American and other minorities. It has a 7 out of 10 California high school rating, so it's not that bad.
There were no recruiters from "elite" private schools. The best students typically go to University of California, either USCD, UCLA, or Berkeley. The next level goes to the other UCs, especially Irvine or Riverside. Many go to San Diego State University, and the rest who continue their education go to community college.
My son loves economics, and got it early in his head that he really wanted to go to University of Chicago. We were pretty poor at the time, as I was changing my career and working my way up in from the total bottom in a new field doing mainly freelance work.
My son worked really hard to get to UChicago. He attended a recruiting event in a downtown hotel he found out about on the internet. I borrowed money and we flew to Chicago for an interview, staying in a sketchy hotel on the South Side. He applied for early admission and got accepted last year, the first student ever from his high school. They offered a financial package that pays almost everything.
His friends who are still at the school said that this year, for the first time, a recruiter from UChicago showed up. It is like my son put the school on their radar.
One thing we noticed as part of the process is that taking the PSAT is really important. It only costs about $20 and it's free in some schools. There might be something similar for the ACT.
My son did well on the PSAT and he got multiple letters from Harvard, Yale, Princeton, Columbia, and other Ivies trying to lure him in. Also from elite private four-year colleges. So there is an outreach by those schools in this way. Often those letters would mention the possibility of almost full support for students that needed it.
I think the Ivies are trying to reach out to poorer students. It's just that it's hard for them to think out of the box created by their own privileged backgrounds and understand how to reach that demographic. But our experience is that they are sincerely trying, and if a student makes the effort, he can definitely get in.
I think one of the biggest barriers is that counselors in schools like my son's or the OP's don't even consider these schools for qualified students (are they uninformed? lazy? incompetent? complacent?). If they pushed harder, both to the Ivies and in guiding the students, it would help a lot.