Anna, I do share your concern about politics being dominated by the elites who have a very narrow view of life and don't understand the concerns of ordinary people. I am also concerned about how political parties are funded and how politicians listen to "big business" / wealthy donors rather than the electorate. Just look at the "cash for access" scandal.
These are all things that need to be addressed.
But direct democracy? No thanks.
The economic consequences of leaving the EU are too complex to be properly understood by the average voter. I firmly believe that decisions such as these should be made by people who have at least a reasonable grasp of the issues.
Of course, economists don't get things right all the time, but when you have such a strong consensus from pretty much all credible economists and institutions that leaving the EU will harm the economy, and the only thing they disagree on is how much harm it will do and how long it will last, it doesn't take a genius to understand that leaving the EU might just be a bad decision from an economic point of view. And yet people voted for it anyway.
Ironically, I think we probably have those powerful elites to thank for Brexit. The Leave campaign was largely bankrolled by multimillionaire Arron Banks. All the owners of the tabloid media (particularly Rupert Murdoch) are pro-Brexit, so their readership was fed a steady diet of pro-Brexit propaganda.
Representative democracy isn't perfect, but it does offer us some degree of protection, and it makes sure that important decisions like this are debated and properly scrutinised.
Direct democracy is all very well, provided you're not deciding on anything more important than who wins X-Factor.