> UK politicians also mislead the public about that.It's worth noting that they didn't just do that in 2016. They were doing it for the entire history of Britain's membership of the EU. During the referendum campaign I talked to quite a few older people (my mother included) who remembered the original Brexit referendum in the 1970s, one that most people outside the UK aren't aware ever existed. It happened just a couple of years after entering the EU, which was done unilaterally by the Labour government of the time and never subjected to a referendum. Many people of that generation say they voted Remain the first time because they were explicitly promised that the EU (or EEC as it was then known) was merely a free trade zone with no ambitions to be more. In that era, without the internet, it was much harder to fact check these sorts of claims and although there were campaigners telling the truth, the public mostly believed the government.
Later it turned out that not only was it untrue but it had never been true, that Labour knew it wasn't true and that the plan right from day one had always been to abolish European governments by merging them together into one superstate. This is often claimed to be a federalist project, but there is no specific philosophy in the EU project as to what rights should remain delegated to the individual states like in the USA. In fact the opening of the EU "constitution" equivalent says that the goal is ever closer union - without limits.
This feeling of betrayal and deception was a big driver of why the older generation went so conclusively for Leave. Most common feedback I got on talking to these people: I haven't changed my opinion from the 70s, but now I know the truth.