There is such a thing as surveys which can capture motivations and reasoning. And then there is the rhethoric of the leave campaign, some of which was obviously untrue back then (the famous NHS line on that bus), and some have turned out to be lies after the referendum (like the optimistic assumptions about a brexit deal). At the very least the voters did not have a clear picture of how the deal would work out.
Your point about "unelected officials" betrays a fundamental misunderstanding of how the EU works. "The EU" or its officials don't decide that much, if anything at all. Most of it relates to being able to exchange goods and services in a fair and safe manner. And even then, these decisions are driven by the elected governments of the member countries, because the "unelected officials" are selected and directed by those.
For example: if a company from country A is providing a service in Country B, using workers from country A, then there need to be certain rules. Labor protections should apply equally, as should professional qualifications and such. Similar things apply to goods traded between countries. If one country lowers its standards, for example for medical equipment, and the other members are forced to accept that equipment, then there is both a safety concern and a competitive concern. That's why so much regulation is coordinated at the EU level - and driven by the member states.
And no, bilateral treaties can't keep up the slack, because 28 member countries would have to have bilateral treaties with each other, which would be impractical.