That the Leave campaign so successfully tapped into the fears of the angry and ignorant makes me worry, a lot, for the US election.
I think one lesson that the Democrats can learn from this is that a negative campaign based on fear and attacking personalities is a good way of alienating the undecided and reducing the chances of success. At no time during the EU Referendum campaign can I remember Remain giving a positive image of a future UK in Europe. The message was all about the how bad things would be if Leave won (they have been right about that so far, but the disillusioned public wanted to hear a more positive message about the future, and that is what Leave delivered, even though their figures and assumptions were just plain wrong). The attack on the personalities of the leading Leave campaigners looked petty and did not resonate at all well with the electorate. And I say that as a rampant Remain voter.
Defending a vision of the status quo is probably much more difficult than delivering a radically different vision of the future that has not yet been tested, but campaigners still need positive messages to give people hope.
Also intriguing is an analysis of issues that resonate the most, as these issues differed among the demographic groups.
The Remain campaign put priority on the economic impact -- the positive impact of remaining in the EU, addressing your concern, as well as the negative impact of leaving the EU -- and that was most persuasive with Remain voters.
And that did resonate with Leave voters, but not as much; the economic impact rated third in their concerns. They put priority on the immigration impact of leaving or remaining in the EU. (They rated as second in their concerns the issue of sovereignty.)
Either pollsters missed this significant disparity in concerns, or the Remain campaign didn't address the pollsters' results to adapt their messages to address concerns -- to provide, as you say, a positive, persuasive response to the top issue for Leave voters.