I would assume in the same way they did when they started voting for UKIP, point being there's always been Eurosceptic parties of one sort or another so "the people" could have made their voice heard and had real influence anytime they choose to, fact is Europe, like you said, had never been a priority for most people in the past but to claim people have never been able to meaningfully influence the situation is disingenuous
IMO.
On that i disagree, the reason why previous governments, of both colours, refused to give us a referendum and happily signed us up to further EU integration (treaties) was because until recently electoral support for Eurosceptic parties was all but non existent.
I don't think even real EU insiders would think the EU is just fine as it is, they may say that but then again most people are reluctant to tell their customers about the cock-ups.
Guy Verhofstadt hit the nail on the head for me when he said Brexit was a "
catfight in Conservative party that got out of hand"