Of course politicians do that, that's a fundamental flaw in democracy. Just look at Trump to see how far and how bad disillusionment with the establishment is in the US.
The demographics make for interesting reading, the majority of people with a degree voted to remain. Also the vast majority of young people voted to stay. It is the less educated older voter who wanted out. I'd imagine they're the least likely demographic to have done a whole pile of research. Certainly the vox pops on TV would lend you to believe that a lot of the leave voters believed the NHS stuff. "i voted leave to save the NHS" was a pretty common thread, as well as anti-immigration stuff of course.
Anyways - it's all a clucking mess now, just saying one vote shouldn't necessarily be the end of it, even the leave side were making a case for a second referendum when they expected to lose narrowly - the petition for a second vote, which has been signed 2 million times, was setup by a brexit person when they were expecting to lose the vote ad has been taken over by remain voters since the vote was lost - which is kinda funny.