Thing is, that figure was still not correct, the £350M was shown to be wrong, the actual number is £180M, which is just over half the figure used by the Exit campaign. And people still went for it because it was "truthy", so it played well to their own preconceptions. Similar to the rhetoric about Eurocrats, when the reality is that the EU has less bureaucrats employed in total than the UK has bureaucrats working in Birmingham, their second largest city.
I think one item that made it very clear that people were voting with their feelings rather than weighing positives and negatives, was Cornwall realizing, after the vote, that they get a lot of EU support, and trying to put pressure on London to match this. Whether or not it will happen, who knows, but after the vote probably wasn't the time to bring it up.
I've said before that I believe that much of the negative feeling towards the EU is from governments all over Europe using the EU as a handy scapegoat, and claiming that any unpopular decision was a result of the EU. This has been going on for 40 years in the case of the UK, which was bound to have an impact. It also doesn't help that people find it difficult to distinguish the EU from the ECHR (a separate organization) and EU related immigration from external immigration (in the UK, the largest number of immigrants are from India, for instance). This is a great example of what the article is saying because it shows that the narrative has been prioritized over the reality, and it's really difficult to dispute a narrative now because it's dispersed, rather than having a small number of sources.