>The student exchange has a longstanding tradition and existed way before the EU shat in its diapers, and during a time where border controls were very strict. And now it thinks it can just cancel that?

Yes, because now the ability to control that program lies with the UE. That's how it works.

>The Swiss people voted for something, it would be unconstitutional and even illegal not to do it now.

Nobody is saying they shouldn't be allowed to do it, it's just that doing it violates an agreement Switzerland made with the EU and there are consequences for that. Consequences that were clear to people who bothered to learn about the issue before voting. This isn't some crazy contrived plot to exert control over the Swiss by the EU, it's just the Swiss having to face the consequences of their actions.