Support for the EU is deeply rooted in a belief that people can't/shouldn't rule themselves: they need to delegate 'complex matters' to elite experts who, of course, only have the people's best interests at heart. The sudden belief that the referendum was only a fancy opinion poll entirely fits with this narrative.
Of course I believe that. Everyone who isn't a rabid libertarian or anarchist believes that, whichever side of political spectrum they come from. I imagine you believe it too to some degree. In representative democracies we don't rule ourselves, we choose the people who will rule us, which isn't the same thing at all.
There's a lot of valid arguments to be had around the process of choosing, the institutions, and what power those chosen should have, but no serious person believes that the best solution for a species that lives in societies is for every member to rule themselves in every particular.
Direct democracy has its place, but that place isn't the UK, which is a parliamentary democracy. It's an arrogation of parliament's responsibility to make decisions based on referenda.
You don't have to be a rabid libertarian or anarchist to think that Direct Democracy has something to offer.