In Europe I think most Universities just rely on the city’s police? Why would an educational facility need their own (assuming it’s close enough to a town or village).
So, apart from the overreaction to protests, larger universities will have their own police departments because they have a large physical footprint and tens of thousands of staff and students whose needs and interests and priorities may not perfectly align with the broader community’s. Students as “gowns” are transitory, young, rowdy, and annoying but often have issues the local “towns” will view as petty and low priority.
A University PD will ultimately report to the University’s leadership, and the University will set their budget and broad mission. Frankly, as a student 25 years ago, at least back then I would generally much preferred to have interactions with the University PD than the local city PD.
Both as a student and as someone that works at a university, it’s always felt that the university police are more understanding of students/young people than the town police. They’re more likely to refer you to student services or explain why you should have reported backing into a car in the parking lot before heading off to class, instead of giving you a ticket for leaving the scene of an accident or arresting you for doing drugs in the library.