None of this is all or nothing. The school could offer catch-up classes. The school and the professor could test and refuse registration in a class for students that are missing the prerequisites. Which means they would need to test at the beginning of the class - and extra work. The school could offer a mechanism for doing that without letting the students stranded with no classes they can register into. etc, etc.

But yeah, of course it's absurd to expect one professor to run this on their own when it's really a school-level issue.