Applicants to a particular school may be widely dispersed geographically, which makes it infeasible for all of them to be at the university in-person to take the exam (appearing in person may be particularly problematic for foreign applicants), and at the same time infeasible to find space and proctors to give exams in every combination of (student location, university). Besides, trusting a local party to administer it would eliminate most of the control you propose to give a university over their own exam. As well, different degree programs at the university would probably want different exams.
Besides that, there's the cost of so much redundant exam creation. Do you think individual universities would have the resources to normalize year-over-year difficult of an entire suite of exams?