Why does CollegeBoard (the organization that does the SAT, GRE, etc.) exist? Why don't universities just give their own entrance exams? It seems such centralization of power in CollegeBoard would be a bad thing for diversity in the universities.
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 difficulty of an entire suite of exams?
There are too many exams to efficiently administer and take if there is one for every university. Combining them into a service makes it easier for everyone involved. If a US student wants to apply to Harvard, MIT, and CalTech, do they have to take one or three exams? What if they apply to ten schools rather than three?
The other answers make good points, but I think they miss some of the most important.
- Having such an exam would not help with any of the status games that US schools like to play. It would discourage applicants, and thus actually be counterproductive in such games (since accepting a low percentage of your applicants is easier if you have more applicants).
- Actually, a big reason that universities pay attention to SAT score is that having an entering class with high SAT scores is one of the status games universities play.
- Effectively, no one at any university has the least interest in creating and grading such an exam. Certainly the faculty would hate it.
- It would require a lot of money (or to use university parlance: "resources") to run such an exam, whereas the SAT is paid for by the students. As discussed above, it would lower a university's status and expend faculty goodwill and time (which is effectively money as well). So, it's a lose/lose/lose.