"What do they need from Harvard to determine if an individual has violated their visa?"

Because the administration has chosen to include define a range of activity which is not obvious from other sources as incompatible with visa status, including membership in certain student organizations.

> Do they not have evidence of a crime?

"Crime" is not the issue, and, no, they don't, that's the problem -- they want information from Harvard with no basis other than the fact that students are on a visa, so that they can use it for fishing expeditions for excuses to deny visa status.

> What do they need from Harvard to determine if an individual has violated their visa? Does the administration not have a list of students on a visa? (Surely they do, given that's their job.) Do they not have evidence of a crime? (Surely they do, otherwise there's no problem. But also apparently not, because they'd just use that.) What's missing?

None of that really matter Harvard is required to report this data to maintain good standing in the Student and Exchange Visitor Program. Failure to do so can result to removal from the program and as such the ability to bring people in on student visas. That is the path they are attacking. Harvard has also been very public about it's refusable to comply with the legal requirements of the program so it's a pretty slam dunk case.

I think Harvard just didn't think the administration would actually follow through.