The first assumption is just flat-out wrong sometimes. Segregation is an excellent example of people acting in a fundamentally irrational manner.
Left to itself, the free market certainly would not have embraced segregation. Segregation was caused primarily by government intervention.
Consider, for example, what happened in the South during the late nineteenth century. Several companies resisted "separate but equal" laws, because paying for separate facilities for different races was expensive. The famous case
Plessy v. Ferguson, which challenged the validity of a Louisiana railroad segregation law, was brought with the encouragement of the railroad companies.
Leaving the free market alone might not solve anything, but government interference will tend to make things worse. So-called market failures are caused, not solved, by the government.