Why Students Pay for Professors' Research

They don't, at least in the USA. My Ph.d. is from Texas at Austin, which is a public institution. During my time as a grad student I had numerous opportunities to 'see how the sausage is made'. Even as a public institution the vast majority of funding (over 80% in the sciences) came from outside grants, such as federal agencies. A professors appointment is also only for 8 months of the year, and they must find outside funding sources to cover the other time.

This is the business model in each department, and it is why promotion to tenure is so focused on publications and grants.

Here's the thing, many classes are taught by (sub)minimum wage temps or grad students. The students often don't get any benefit from the researchers.

It would be nice if colleges published how much of your tuition dollar actually went to the teachers - a bit like some charities do.