To grossly oversimplify the point, I think you could classify a lot of writers as being Shakespearean or Shavian in the sense that--

--Shakespeare generally likes to talk about (and is probably best known and remembered for) tackling "the big questions" of human nature, the psyche, emotions, the human condition, the nature of love, power, etc. It's part of why "Hamlet" is considered by many to be his jewel amongst jewels and the character of Hamlet's arguably the most influential in all of English literature--he's ALL ABOUT taking on these big, issues and trying to ESCAPE his state of idleness (even though Freud argues Hamlet's not really idle, and I somewhat agree with tat, but I digress) by way of reasoning his way through these immense ethical, theological and philosophical dilemmas. He has the most lines of any Shakespeare character and his lines are the most widely quoted of any Shakespeare character. He's not just all idleness and imagination, he's trying to be actively putting his thoughts to use, which is ultimately what Shakespeare does throughout his works.

Dostoyevsky does this a lot too (Prince Myshkin and all three Brothers Karamazov try and use their ideological platforms to solve their problems) and the same can be said of writers like Dante and T.S. Eliot.

--By contrast, a Shavian writer, like Shaw himself, will tend to be more concerned with tackling social issues. You can't tiptoe through a Shaw play without some practical discussion of class and power structure, capitalism vs. socialism, democracy vs. alternatives to democracy, critique after critique of religion and especially religious institutions, gender roles...and often all in the same play.

Writers like Orwell and Huxley are in a similar vein--different politics, sure, but they're still arguing for some forms of social reform and against political ideals they see as damaging or even dangerous. If we wanted to really run the gamut here we could say that this holds true for racial writers (Langston Hughes, at least in the first stage of his poetic career, is very much concerned with furthering Black Americans, he's in no way just being imaginative or idle) and feminist writers (Rebecca West is a good example) and definite political writers (Voltaire, Swift, Eugene O'Neill and Ayn Rand are all definite political writers...they're very, VERY different in terms of what they're arguing for and chances are you're not going to like or agree with all four or maybe even two out of the four, but you can't argue they're celebrating idleness or just playing around with their imagination, Swift's fantasy worlds are all made for a carefully-crafted satirical and political point) and so on.

The "trap authors" here are Oscar Wilde and Bertrand Russell (an odd pairing, admittedly.) Both notably wrote things "celebrating" idleness.

But where Russell wrote "In Praise of Idleness," no one would ever DARE saying that he was chiefly concerned with that or that that was his ideological point on writing--he was a huge activist, and his writings are definitely concerned with both political issues as well as grander ideas of logic and reason and how that might be applied.

Wilde's probably the closest you'll come to someone who fits this, his "art for art's sake" quote's easily one of his most famous (and as that's Wilde, that's saying something.) And I think it's fair to say that at least some of his works do deal with decadence and idleness, positions his male protagonists in particular are pretty famous for. Wilde still's far too witty to leave it there and does something with his idleness, he makes his idleness "useful" and a platform to talk about and argue for and against things, but still, it is fair to say that if there's any author who's a counterexample here, it's probably Wilde.

Still, I don't think he's a fully counterexample, and I think literature SHOULD be useful...not necessarily in 1-to-1 teachings of morals and ethics and all of that (that tends to lead to overly-patronizing and pretty boring reads anyway) but you can still create a work of literature that's "useful" and argues for and against things and still layer it enough and include enough nuance that the reader's allowed to draw their own conclusions and, as a result, literature DOES end up serving some social utility, it's not just "art for art's sake" (which I'd again say, in my own personal taste, is the kind of art that I usually end up liking the least or outright disliking...If you want to be experimental with literature and art, that's fine, but don't just take a few eggs and hurl them randomly against a canvas and call that a masterpiece on par with Picasso if only I "interpret it" according to your "feelings" or my "feelings," and definitely don't write a one-word "poem" and claim it's as good as any ode or sonnet. THAT is art for art's sake taken to the extreme, and it's not a good or, indeed, useful extreme at all...and I'd think most people would prefer a Picasso or Monet to an egg-soaked canvas--and there's a reason why.)