Denethor may have the strongest case.
The Denethor of the film has no redeeming features. From the moment we see his disheveled and haggard face peering out at us through bad hair, we dislike him. He rants, he spits, and if you bother to think about it you see pretty clearly that he is batshit insane. He not only denies help when it is offered, he refuses to request aid from the only people that could possibly save him -- the Riders of Rohan. He sends his elder son on a power grab for the Ring and his younger son on a suicide mission, and he does it all while ordering Pippin to sing. Oh, and he's a messy eater. A slob, really. The Steward of Gondor is a slob.
When he finally turns himself into a staggering bonfire, we hear Sir Ian intone, "So passes Denethor, of the line of Ecthelion," but what we're really thinking is, "That GOD that stupid-ass dude is dead; let's see more frickin' mastodons!"
Look now to the Denethor that Tolkien depicts for us.
Then the old man looked up. Pippin saw his carven face with its proud bones and skin like ivory, and the long curved nose between the dark deep eyes; and he was reminded not so much of Boromir as of Aragorn. -- "Minas Tirith"
Denethor is physically supposed to remind us of Aragorn; he has strong Dunedain blood, like Faramir and Prince Imrahil, but unlike Boromir, his eldest son. But if he is of the Dunedain, then why is he an "old man" at the age of merely 78? We learn in the Appendices to LotR that Denethor aged early because of his frequent use of the palantir, which unbeknownst to him was utterly under the command of Sauron, and because of this whenever Denethor looked in the palantir he was allowed only to see those things which Sauron allowed him to see. Thus, he saw armies growing always and beyond hope of defeat. And although he felt that there could be no victory against this foe, yet he was determined to fight it, and in this his body and mind fell to despair.
Many fans of Jackson's Return of the King recall the song Pippin sings for the Steward; a song the actor wrote himself the night before the scene was shot, so rushed were the scriptwriters at that time. In the novel, Denethor does reassure Pippin that his happy hobbit songs would be welcome:
'And why should such songs be unfit for my halls, or for such hours as these? We who have long lived under the Shadow may surely listen to echoes from a land untroubled by it? Then we may feel that our vigil was not fruitless, though it may have been thankless.' -- "The Siege of Gondor"
But in fact, Denethor does not command Pippin to sing at all. Instead, he questions him intensely on the people of Rohan and especially Eomer, though he seems to already know all the answers to his own questions. This is because he has been looking west with the Palantir, and knows much that has gone on there already. But my point here is just that, rather than getting fruit juice all over his face and listening to questionable songs while his son Faramir goes off on a suicide mission, Denethor actually spends this time gathering intelligence on a possible ally.
Oh, and it is worth pointing out here that in the novel the forces of Gondor still held their side of the river in Osgiliath when Faramir was sent to reinforce it, so although the mission was certainly perilous, it was not the certain death it is made out to be in the film. When Faramir and his men retreat to Minas Tirith, Denethor orders a sortie from the walls which allows Faramir to be brought -- injured -- inside.
Because Denethor may look old, and he may spend a lot of his time sitting around planning, but he is also resolved to spending the rest of his life in battle with Sauron.
'Or why should I sit here in my tower and think, and watch, and wait, spending even my sons? For I can still wield a brand.'
He stood up and cast open his long black cloak, and behold! he was clad in mail beneath, and girt with a long sword, great-hilted in a sheath of black and silver. 'Thus have I walked, and thus now for many years have I slept,' he said, 'lest with age the body should grow soft and timid.' -- The Seige of Gondor
Denethor sleeps with his sword and armor on so that he will never allow his body to grow too weak to fight. Can you imagine a bigger contrast to the raving kook of Jackson's film?
But I'm not done yet, because the strength of Denethor was not in his body. It was in his mind. His will is stern enough that he can do mental battle with no less than Gandalf himself, though against Gandalf the White he cannot hope but prove the weaker. This very interesting scene, in which Denethor and Gandalf both literally read each other's mind in a kind of psychic struggle, takes place in the chapter "Minas Tirith."
He turned his dark eyes on Gandalf, and now Pippin saw a likeness between the two, and he felt the strain between them, almost as if he saw a line of smoldering fire, drawn from eye to eye, that might suddenly burst into flame.
Denethor looked indeed much more like a great wizard than Gandalf did, more kingly, beautiful, and powerful; and older. ... And then [Pippin's] musings broke off, and he saw that Denethor and Gandalf still looked each other in the eye, as if reading the other's mind. But it was Denethor who first withdrew his gaze.
The Steward of Gondor is indeed mad by this time -- but his madness is a subtle madness that none save Gandalf and Pippin understand, and they only at the end. Denethor has used the palantir to see Sauron's ever-increasing strength, and he has decided the fight is unwinnable, but still he resolves to fight it. He comes to see Sauron as his own personal nemesis, and he is willing to sacrifice anything and everything to defeat that Enemy, even if it requires sacrificing his own sons. He lights the bonfires that send word north. He sends the Red Arrow to Theoden to remind the King of Rohan of his obligation. He calls up every available man from the southern provinces of Gondor. But few come to his call. Denethor only finally loses hope when he perceives that Faramir is going to die, and when the palantir shows him a fleet with black sails coming up the river from the south. This is actually Aragorn's fleet, but Sauron does not allow Denethor to see that, and the Steward knows he cannot fight an army of Southrons in addition to those forces already present. And of course, he knows nothing of Frodo's mission, so has no cause to hope, unlike Theoden, who learns of that small hope from Gandalf.
And that may be the best way to see Denethor, after all this: he is Theoden, un-saved. He is 'What if Theoden had not been freed from Grima's whisperings, and had fallen completely into despair.' Still a great man, indeed one of the greatest of Middle-earth ... but no match for Sauron, the Enemy he had chosen to oppose single-handedly.