Right up to the referendum (June 2016) most opinion polls were showing a vote to remain

I fully understand why you say that, and it was indeed the message the press and other media were saying.   However, this link from YouGov is worth reading. I believe, also, that the media did not pay anything like enough attention to the range of results within the confidence intervals, just going with the central prediction. Understandable, to try to keep things simple, but misleading.