The problem isn't in the Schengen Agreement, the problem is in the free travel right between EU states. Schengen makes it smoother, but it's ultimately the Maastricht Treaty which makes it a right.
The idea behind refugee distribution would be that refugees would get a refugee status from the EU, and then be distributed over the member states. The idea would be that this would deter economic immigrants from seeking refugee status, as they couldn't count on getting refugee status in the country of their choice. Real refugees on the other hand are fleeing from a country, and any safe country would be fine to them.
Eastern European countries did point out that this wouldn't deter those economic immigrants, as they could just wait until they receive EU citizenship and then still travel to their intended destination. While this is true, it would take years and that is likely a deterrent.
Schengen is irrelevant. It makes border crossings easy for tourists, truckers and business travel, but we were talking about migration. That's a far more permanent sort of border crossing.