Your take is a bit like This explanation of Brexit has gone viral.

A part I think you're missing is that the tabloids having been using the EU as an easy way to rile up their readership and deflect frustrations regarding austerity etc. for years.

In doing this they allowed UKIP a very loud voice whilst at the same time ignoring the fact that UKIP don't actually understand how EU democracy works (well you wouldn't if you don't actually get involved) or what trade deals involve.

Misunderstandings, half truths and outright lies become common knowledge even amongst politicians.

Then comes the reverse physcology (victim mentality maybe a better phrase) of claiming the loudest voice, as a result of constant tabloid coverage, is the minority voice that the establishment doesn't want you to hear (even though the tabloids are a major part of the establishment).

Then people vote according to bullcrap.