In my opinion, it should be illegal for customers to try to avoid vat by offering [sometimes demanding] to pay cash
I quite agree with the sentiment. However enforcement ?
I am amused that a lot of this discussion seems to assume HMRC are some kind of boy scout movement armed with a smile and nothing else. They already have powers so vast and far-ranging that they could target tax evaders head-on.
If they wanted.
First off, when they find any evidence of tax evasion - either in a business or an individual, they can demand an estimated tax payment. If the subject disagrees, it's up to the subject to disprove the assessment.
If they suspect fraud, they can demand documentation going back to the year dot - none of this "6 years" nonsense. (HINT. If they do, never supply them - apparently being able to do so makes you "more" guilty". BTDTGTTS).
But none of these powers mean a jot, when the head of HMRC goes on business lunches with companies under investigation, and returns telling his subordinates he's "sorted it all out" and there's no tax to pay (looks at Vodafone, inter alia).