The problem with direct democracy, where the people direct the government to do things, is that you require someone to enact the policy that the people vote for. There is no tradition of referenda in Britain, rather they go against our tradition. Whenever they have been held before, they have gone like this: the government, or a regional government, wants to do something, and they therefore ask the people to give them their permission to do it.
The Brexit plebiscite did not follow this convention at all. Instead it went like this: the government wanted not to do something (leave the EU), and asked the population to tell it that they did not want it to do the thing. Instead, the public told the government that they wanted it to do what it didn't want to. Usually, when a government does not want to do something, it has the good sense not to call a referendum on the matter. However, David Cameron was a foolhardy and inexperienced politician who recklessly called a referendum with no plan if it did not go his way.
In effect, Britain elected a policy (Brexit) without a government with the will and the means to accomplish it. It may have been possible, if British leaders were good at their jobs, knowledgeable and wise, but they actually embody the very opposite of these characteristics. You need a government in power who want to do what the people want or else direct government does not work.