Come to think of it, the only papers which their readers would miss are the ones which have have managed to establish their names and the word ‘reader’ as a social type: which is to say the Guardian, the Telegraph and the Mail.
John Band argues that this is also true of the Sun:
Surely 'Sun-reader' and the Sun also fit alongside Guardian, Mail and Telegraph?
Certainly, people do use "Sun reader" as a social type. But the really interesting question is whether anyone considers themselves a Sun-reader, and I think this is what's doing the work here. (As a very rough check, I compared the Google hits for "I am a [paper] reader" - Guardian most common, Sun a couple of thousand less, but quite a few of both were people either putting it on for argument's sake or indignantly denying it. Obviously, the huge Guardian web presence will distort that.)
People who read the Guardian often do identify as Guardian-readers and other people also pin it on them. This is true, although I think more weakly, of Telegraph or Mail readers. But there is a gradient here - I think you're slightly more likely to self-identify as a Telegraph or Spectator reader than you are to be labelled as one. For the Mail, that's more like evens. (The reductio ad absurdum would be the Daily Sport, which would almost certainly be an insult.)
For the Sun? I'd put it at 80% labelling to 20% self-identification. Why is this important?
Well, you can define a following - Sun readers, Worcester Women, whatever - and use this to sell advertising or push your own influence. In the first case, what matters is that you can define your own readership well enough that advertisers think of you as a way of reaching them. In the second, it's that politicians are willing to believe that Sun-readers are a thing. Note that this involves a willing suspension of disbelief. If you can count the C2s among your readers, your media sales team can throw this at advertisers. If you are ideologically congenial to politicians, so that they're willing to believe in Sun readers, you can exercise power. In a limited sense, if you can render your audience legible as a group, you can turn this into money or influence.
But this only goes so far. The key distinction is what happens when you need them. People who identify themselves as Sun readers will turn out. People who are identified by marketers as Sun readers will read something else in their tea break. And there's an odd recursive quality to this - if you really did consider yourself a Sun-reader, what on earth would you be doing identifying yourself as a newspaper reader? What could be less in keeping?
To put it another way, imagine someone who is acting, trying to pretend to be a Sun-reader. What could be more obviously fake than brandishing a copy? You would need to work on the rest of the act first, and only then have one casually lying around. If you wanted to pose as a Guardian reader, you'd want to be seen reading the damn paper.
News International spent a lot of time and effort trying to create an identity for NI-consumers (there being not much difference between the target demographics for the Sun, the NOTW, and Sky Sports, and a hell of a lot of cross-promotion). Of course, so do all media products, even Mobile Comms International and Elevator Week. Some would deny it (The Economist), some would boast of it (The Face). Some are more successful than others.
But I would argue that rather than observing what its customers wanted and marketing it back to them, or deciding what they ought to want and persuading them to want it, NI's modus operandi was to observe what its customers did, and then market that to its other, upstream customer base - advertisers and politicians.
If the Sun called for a demonstration against the Leveson inquiry, would anyone go?