///A dedicated gov.dept. to police this would more than cover its cost 100 x over ///
Really, do you know this?
To give a similar (but smaller example), our town introduced residents parking a couple of years ago. You would think that be charging £25 for every permit and enforcing £60 fines for people not complying that it would be revenue neutral. In fact it loses tens of thousands of pounds per year.
A government department (and a computer system) will cost many millions to run and maintain.
The point with every law is that it is pretty easy to break. It is illegal to drive at over 70mph, but cars are made to exceed this speed. You don't have to prove that you will only drive at 70 when you buy a car. BUT, if you are caught breaking the law, you should and will be punished.
Fraud takes place in pretty much every country - it will never be stopped. Whether it is benefit fraud, banking fraud, insurance fraud or whatever - it happens all over.