Demand for exclusive institutions by the elite won't go away if you ban legacy admissions. Expect some new unaccredited institution to open up, and that's where they'll attend.
The rich don't need diplomas, if they can just take over the family's business. In fact they don't even need a job at all. But they may need a few smart people for important positions in said family's business. In an institution composed of both rich and smart people, connections naturally form, so the rich get the smart people they need, and the smart get a good job.
Separate them what will happen is that your future boss will not be as well educated, and will probably hire some of his rich friends instead of his smart friends. And the smart will have a harder time finding the jobs they deserve.