//Do you honestly believe that with a victory that narrow the Leave side would have won without anti-immigration voters?//
So what if it did take anti-immigration voters to swing the poll. They are as entitled to express their views through the ballot box as anybody else. Does being of an anti-immigration persuasion somehow disbar one from voting? This country was taken into the EU without a plebiscite. It's often said that those voting to leave it did not know what they were voting for. The same is equally if not more applicable when the electorate was asked to confirm its support for membership in 1975. Nobody knew 45 years ago that the trading organisation that they supported would have ambitions to become a federal state; nor did they know that the bloc would expand to encompass many USSR satellite states of many, quite frankly, don't have a pot to pee in. And they certainly did not know that "EU law" would reign supreme over their own Parliament's legislation. If you believe they were told all of this, have a look at the pamphlet the government produced ahead of that referendum:
http://www.harvard-digital.co.uk/euro/pamphlet.htm
All politicians of every persuasion mislead the public. It's what they do and it's about the only thing of which they demonstrate any capability. It's up to the electorate to see through their deceit and cast their votes according to what they know and believe, not what they're told by politicians. That's what it did in 2016.