Yes, this is probably the correct thing to do. Unfortunately, most universities are too cash-strapped for such a thing.
They're paying uni professors/lecturers to do admin work that they could pay a fraction of the same amount to have done. I know that in practice it's more complex than this - for example here it only works because the lecturer is not actually being paid by the hour; they're still under-utilising their lecturers.
Create a position (or probably department) for quality control. Automatically send "you plagiarised, warning" letters to the student and the person who paid for the course if the submitted work receives a score above X threshold for plagiarism (this threshold to be chosen by an inter-university body and preferably standardised across universities).
Obviously a student would be able to appeal and have proper analysis done by a department member. A second warning without appeal would get a student a P grade for "admitted plagiarism".
This way the teacher just teaches, just marks the work submitted and the administrative duty of notifying students that their plagiarism has been spotted is separated and codified.