If someone notices that morgage defaults are uncorrelated, and new instruments that exploit that lack of correlation are invented, investors will be more likely to buy securitized morgages, so the rates are bid down and they become cheaper for homeowners. But that in itself should not cause defaults to be more correlated.
In the event, the models for correlation were off, and defaults were more correlated from the start. But that’s not a problem with public/private knowledge or anything; they were just bad models. If the prizing had been done correctly to start with, I see nothing paradoxical in assuming that introducing CDOs would lead to persistently lower morgage rates—the lack of correlation doesn’t stop being real just because people believe in it.