I suspect that if any leave voter had asked any EU politician a question about a policy that affected the UK they would have been given a reply at least close to the truth.
Yes, but few Leave voters were that engaged - or would have understood the answer. Remember the level of "debate" was "take back control", "brown people out", and "bent bananas".
Many (many !!) years ago, DW started studying law, and had some (ing expensive) textbooks. Idly leafing through them one day, I caught reference to EEC (as was) law over UK law. I can't recall the case, but pretty soon after the UK signed up to the EEC, a case came to the UK courts where UK law (it might have been sex discrimination) was at odds with the EEC regulations. After judgement, appeal, and the Law Lords, the result was the notion that as Parliament had ratified the EEC treaty, then it was a clear intention that Parliament intended the UK to comply with the said treaty. And in instances where UK law diverged from EEC rulings, in the absence of Parliaments ruling the UK courts were required to interpret the case assuming the EEC version of the truth.
Or at least that was what I took away from the article. The section of the book it was in was outlining the sources of law in England and Wales, with a note that the evolution of Parliamentary Sovereignty meant that the UKs constitution allowed for quite dramatic changes (i.e. joining the EEC) with no need for adjustment.