Why does the public/government sector have some control over the businesses in the private sector?
The simple answer to this question is that this control represents the will of the people. We elect people who write the laws that control the business sector, and at least theoretically, they do so at our behest. Of course, the situation is far more complex.
Let's turn this question around for a moment and ask what happens when a government exerts no control over the private business sector. This is essentially unfettered capitalism. In this world, work environments would not be safe. Supply and demand alone would dictate wages, so that in times of high unemployment, for example, businesses could pay people $1.00 a day. There would be no guarantee that any product sold was safe. And advertisements could outright lie about the products being sold. Manufacturers could pollute the environment with no restraints whatsoever. A person discriminated against in employment would have no legal recourse. I do not want to live in that world, and I do not believe most other people do, either.
Government is meant to protect all of its citizens. That is its job. Government regulation of the private sector flows from that responsibility. As you can see from the examples above, without government regulation of business, the consequences for many people would not be good. And we know, historically, that this is the case. Prior to the government exerting control over businesses, people sickened and died on the job, many worked for what were essentially slave wages, the environment was tainted beyond belief, and there was no consumer protection. With government control over some aspects of business, lives have been saved, people get higher wages, and consumers can reasonably count on the safety and efficacy of the products they purchase. It is a better world for all citizens, and it is evidence that government is doing its job.