The problem with encouraging people to study at 'local' universities is that whereas there are league tables for institutions as a whole, these can mask differences between the quality of courses in individual subjects, so the local one might not be the best for the subject an individual student wants to study. Also, things like options and even core modules vary depending on staffing, so even if a university offers an excellent course in, say, History, it might not have modules in medieval French towns (or whatever), which might be the thing that interests a particular student.
This would mean, as always, that students from wealthy backgrounds will have choices that are denied those from more 'ordinary' families. Also, whereas social life is important to many students, it is, arguably, more important that they mix with people from different backgrounds, and this is less likely if they stay put. Those going to (or staying in) better off areas will make contacts who won't be as likely to go to poorer areas to study. Similarly, many universities work with local companies for placements and research collaboration, which will reduce chances for local students to find work outside of their own area. People wanting to study Art, for instance, would find it far more difficult in an area with few galleries to visit than somewhere with one on every corner.
I think that to some extent it is true that less well-off students do stay at home, but their experience is not really the same as those who go away and use the experience as a bridge between childhood living with parents and self-sufficient adulthood.