Why do researchers need universities?
If I'm understanding the answer correct, professors need funding to do research, but once they get it, the university takes some of the grant as overheads (to pay for office space, electricity, etc), and the cut the university takes is substantial


The number of services provided by a university is substantial, of which space and electricity are the least of them.

Maintenance, janitorial, and IT services - machines break, rooms get dirty, and technology has inexplicable problems.  Universities have infrastructure in place to deal with all of this.  

Access to literature - Universities maintain subscriptions to the journals, standards, and other references a researcher needs.  

Laboratory equipment and specialists - Most scientific research requires specialized equipment, which can cost anywhere from a few hundred dollars to several million.  Universities can afford to maintain such equipment, and the expert technicians needed to run the equipment at maximum efficiency.   

Other researchers - A university department provides ready access to and familiarity with other experts in your field for potential collaboration.  Additionally, it provides a way to find potential collaborators from other fields for the times when you find your research leaving the bounds of your expertise.   

Students - an unfortunate amount of research is tedious.  A university provides a ready framework for delegating simple but time-consuming work to students so that a researcher can focus on the parts of the research that only they can do.  

Contacts - Universities maintain contacts with businesses, governments, and other research institutions that can open more doors than any one researcher could do on their own.   

Reputation - Being a member of a respected university means that people who trust that university will afford some measure of that trust as well.  

Legal services and other expertise - While it would be nice if research could exist completely separate from the outside world, this is not the case. Universities have systems set up to support and advise their researchers when problems arise during their research.

Many of these things could be aquired by a solo researcher, of course. But doing so would cost time and money, likely amounting to more than a university takes in overhead. And things like contacts and reputation are difficult to purchase.