"One side" both sides outright lied. So have to disagree on at least that point.

As I keep being told the UK is a 'representative democracy', it is generally your MP who decides the specifics, as the voting electorate it's your job to decide which strategy should be followed. This is why manifestos are presented and that they are not complete covering every policy. The EU also made it abundantly clear (as they threw their toys out of the pram) that the UK could not 'cherry pick' parts of the EU, they were either in or out. There was no option of remaining part of the single market so it would be silly to offer that to the public.

Yes or No, was a legally non-binding way to advise government. What the government did with that information was their own effort. The referendum was sound. That someone lied should not preclude us from voting on things, because all politicians lie... except perhaps in wales where it might actually be a criminal offense for politicians to lie soon(tm)...

There actually was a deal with the EU, several of them, however, the public then elected Boris Johnson deciding by electoral mandate that they wanted out and leave meant leave, not half in half out, just fuck 'em. It's rather sweet to pretend that there wasn't a general election before we actually left. There was a chance for the public to opt towards trying to remain in the single market, whether via Theresa May's deal or by electing Labour.