Amongst the many, many problems with winner takes all, 'once in a generation' style constitutional referendums is the fundamental fact there's zero incentive for the actors to campaign with any level of honesty.
In retrospect, it was probably strictly rational to be as dishonest as practically possible; because there never was any sanction for 'cheating'.

Consider the most basic rationality game - the prisoner's choice - and then remove the punishment for defecting. The game breaks.
I'd suggest whichever side presented their case most honestly was destined to lose.

Leave's offer - "Leaving the EU will be painful but the benefit in self governance/sovereignty will be worth it" does not win the referendum. Hence, "there are no disadvantages, only advantages".
Remain's offer - " EU is an imperfect, flawed institution that's in need of reform but it's vastly better for the UK to be inside than out" doesn't win.

But, Remain's offer was far closer to the truth than Leave's (which was almost entirely fantasy) so Remain lost.

¯\\_(ツ)_/¯
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