This question is based on the pie-charts shown in this link, such as this one for the University of Michigan.
In most of these charts, around 5% of the revenue comes from gifts. What are these exactly?
How do hospitals and clinical services play such a critical role for the revenue?
Gifts are donations, typically from former students but they could come from anyone. That's just how higher education works in the US, and the tradition of making these donations is a big factor in how some US universities have become very wealthy (some of the donations are spent each year, but many are added to the endowment). The donations qualify for a tax deduction, so there's an element of government subsidy as well.
Hospitals and clinical services can loom large because university teaching hospitals are huge and medical services are very expensive. (You can have thousands of employees, some of them very well paid, and elaborate equipment.) However, this is really not relevant. The medical revenue is typically paying for the hospital activities and medical education, but not used to subsidize the rest of the university. Many universities do not have hospitals or medical schools.