Earlier quoted context omitted.
It was my understanding that the effectiveness of direct democracies has been inversely related to size of population throughout history... So, specifically in the context of discussing direct democracy, size does matter.
That is a good point, but, how many of those democracies took place in the era of fast communication by locomotive, telegraph, email, SMS, the long distance phone call, Skype, and others? Surely there is a difference between the Roman Empire being stretched and sending messages by foot and FDR addressing the US by radio.
Even in the relatively recent founding history of the United States, the Federalist Papers discuss and try to address a variety of issues which can lead to this outcome.
And ultimately, I think this is what we see with the Brexit outcome and the current Trump campaign in the US. These aren't "new" concepts...throughout history we can see similar outcomes for similar events with similar reasons.
You raise an excellent point though, which is with the ease of communication in the modern era, why don't we have a more informed public? This is something I have thought about trying to tackle with a side dev project...but the issue is complex and difficult to address directly.
Side note, I believe the Roman Empire was a Republic, not a Direct Democracy :)