Rhi,
But as I've just said the DM subverts democracy. It doesn't publish news, it publishes fake news. The people that read it do so because this is the kind of news they want so you can't expect regulation on the part of its readership. They want to hear that it's all the fault of immigrants/Muslims/brown people, or that coffee/bacon/anoraks give you cancer.
What part does it play in a healthy democratic society?
Very little - it's pretty much anathema to that. But here you're describing the liberal paradox - if you believe in freedom of speech, then on what basis would you deny it to others, however filthy, bigoted, dishonest etc they might be? The moment you say "ban it!" you weaken the very thing you seek to protect. If the DM, why not The Daily Express? And if the Express, why not The Sun etc?
And if nonetheless an authoritarian gov't gets in and says, "Ban the Guardian" what defence would you have?
That's why banning even some freedom of speech is so problematic - the classic example being that you can't run into a crowded theatre and shout "Fire!" because real people are likely to be hurt that way. But next we have phrases like "likely to incite violence" etc. Likely according to whom, and where should that line be drawn?
Just look at Theresa May now and the heavy hints about increasing the Gov't's right to look at private e-mails under the guise or reducing terrorist threats. Do you really think Trump wouldn't close down CNN or The New York Times if he had the power to do it?
That's the irony - that when you trade privacy for security eventually you deserve neither.