Welcome to the board!
Yes, absolutely boarding schools want middle class kids! Pretty much every school has a minority but significant percentage of middle class kids. It varies, but my sense is that the schools see them as an important part of balancing out the demographics. Having only rich kids and full ride kids isn’t ideal. The way my kid describes it at his school, 20% are conspicuously rich, 20% want to be in their orbit, and everyone else is normal and very friendly. But truth is, often you can’t tell who is wealthy- lots of really down to earth people have money. There’s not a lot to spend money once on the boarding school campus. The main ways it shows is vacations and clothes and getting food delivered to campus. But not all wealthy families value those things.
Conventional wisdom is that boarding schools are more generous with financial aid than colleges. So the sweet spot is finding a school that fits a kid’s learning and living style, that can give you enough aid to make it do-able.
There is the question of getting admitted - these schools can be very competitive. There is a really great thread here right now on one dad’s learning curve on finding a school with aid. Start there for your research. Don’t limit yourself to the most prestigious or well known. Lots of really great schools out there. Let us know what questions you have.