Why do some people believe that a mixed economic system solves basic economic problems?

It is because both the private sector and public sector have a say in answering the basic economic questions, thus, there will be a degree of high efficiency (due to the private sector involvement) and social welfare too (due to the public sector involvement).

It allows the Government to intervene when the economy faces market failure. The mixed economic system allows markets to operate freely until it fails to allocate resources efficiently, after this, the government agencies, such as the EU Competition Commission, are able to correct for these failures. If this had been a completely free economic system, Government agencies would not have the means of power to be able to intervene.