Be very careful here. Total absence of competition means lethargy and boredom that can kill your kid's spirit.
Third-string private LACs have plenty of kids from rich families with all the negatives that come from wealth with none of the drive and intellectual curiosity more typical of competitive schools. Basically "play hard" mentality without "work hard".
The OP said she wants colleges that do not attract wealthy families. These can be found among private SLACs. Many of them do not have the kids of rich families. OP, visit the schools. You will be surprised that many are full of middle class kids who are driven by the fact that they must work upon graduation.