This prevents the situation from getting out of hand before it completely descends into violence.
That's it. That's the sum total of what democracy gets us.
Analyzing critical ideas is not now, and wasn't ever part of the democratic process. Cicero was complaining about citizen's lack of analyzing critical ideas (except when directly hurting them in the wallet) in 70 BC. You can read about leaders from the French revolution lamenting the same, and even in cases where only limited self government exists, like today in Bahrain, you find most analysts complaining that the large majority of the people who have influence (usually local landowners) do not think critically at all.
What has happened is simple: the previous government was pro-Europe, couldn't get Europe to commit to basic concessions that would make British lives easier, and the imposed austerity made a lot of people's lives worse in a very concrete and visible way. So they were voted out.
Now you can argue that they were right, that it wasn't their fault that this happened, and I might even agree with you. But as the government, I do understand, this is not good enough.