People often say LOTR is a story about hope. (I'm reminded of it because someone said it in the notes of my Faramir post.) And that's true, but it's not the whole picture: LOTR is in large part a story about having to go on in the absence of hope.
Frodo has lost hope, as well as the ability to access any positive emotion, by Return. He is already losing it in Towers: he keeps going through duty and determination and of course Sam's constant help.
For most of the story, Sam is fueled by hope, which is why it's such a huge moment when he finally lets go of the hope of surviving and returning home, and focuses on making it to the Mountain. To speed their way and lighten the load, he throws his beloved pots and pans into a pit, accepting that he will never cook, or eat, again.
When Eowyn kills the Witch King, she's beyond hope and seeking for a glorious death in battle. It's possible that in addition to her love and loyalty for Théoden, she's strengthened by her hopelessness, the fear of the Nazgúl cannot touch someone who's already past despair.
Faramir is his father's son, he doesn't have any more hope of Gondor's victory or survival than Denethor does, he says as much to Frodo. What hope have we? It is long since we had any hope. ... We are a failing people, a springless autumn. He knows he's fighting a losing war and it's killing him. When he rejects the ring, he doesn't do it in the hope that his people can survive without it, he has good reason to believe they cannot. He acts correctly in the absence of hope.
Of course LOTR has a (mostly) happy ending, all the unlikely hopes come true, the characters who have lost hope gain what they didn't even hope for, and everyone is rewarded for their bravery and goodness, so on some level the message is that hope was justified. But the book never chastises characters who lost hope, it was completely reasonable of them to do so. Despair pushed Théoden and Denethor into inaction, pushed Saruman into collaboration, but the characters who despaired and held up under the weight of despair are Tolkien's real heroes.
(In an early draft of Return, Frodo and Sam receive honorary titles in Noldorin: Endurance beyond Hope and Hope Unquenchable, respectively. Then he cut it, probably because it was stating the themes of the entire book way too obviously, because this is what Tolkien cared about, really: enduring beyond hope. Without hope.)
Also, people who know more than me about the concept of estel, feel free to @ me.
I get what you're saying, and I mostly agree with it, but I don't think "hopeless" is quite the word for it, and is probably at least partially why Tolkien axed the Noldorian titles. As you mentioned, Tolkien had two different concepts of "hope": estel and amdir.
Amdir is practical hope. It's saying, "ok, we do this, that, and the other thing and we'll probably be ok." It's the strategizing, action-oriented, mindset that most modern folk would think of when they hear "hope". Boromir was very much a man of amdir. Amdir is why he was a good general, why he jumped to plow snow with nothing but his arms on Caradhras with Aragorn so they could all make it back down the mountain. Amdir is good, but it is also fallible.
Estel is... not that. Estel is what Tolkien called "high hope", the enduring hope that the world will turn out right in the end, regardless of what that means for you personally. Estel is saying "I will do what should be done, what must be done, even if it kills me." I guess the closest word would be "faith", but I hesitate to call it that, given the extremely religious connotations that word can have, and, as devout as Tolkien was, he didn't really use estel quite like that. Estel was not a thing that required belief in, or even knowledge of, Ilúvatar or anything like that.
Frodo and Sam are running on pure estel by the end of RotK. Well, everyone left of the broken Fellowship is by that point, tbh. So you could say that their hope for their own personal selves was gone out the window by that point but that's not quite the same thing as being hopeless.
Sam even has a moment after ditching his pots of his own estel and fallen amdir arguing with each other:
He could not sleep and he held a debate with himself. ‘Well, come now, we’ve done better than you hoped,’ he said sturdily. ‘Began well anyway. I reckon we crossed half the distance before we stopped. One more day will do it.’ And then he paused.
‘Don’t be a fool, Sam Gamgee,’ came an answer in his own voice. ‘He won’t go another day like that, if he moves at all. And you can’t go on much longer giving him all the water and most of the food.’
‘I can go on a good way though, and I will.’
‘Where to?’
‘To the Mountain, of course.’
‘But what then, Sam Gamgee, what then? When you get there, what are you going to do? He won’t be able to do anything for himself.’
To his dismay Sam realized that he had not got an answer to this. He had no clear idea at all. Frodo had not spoken much to him of his errand, and Sam only knew vaguely that the Ring had somehow to be put into the fire. ‘The Cracks of Doom,’ he muttered, the old name rising to his mind. ‘Well, if Master knows how to find them, I don’t.’
‘There you are!’ came the answer. ‘It’s all quite useless. He said so himself. You are the fool, going on hoping and toiling. You could have lain down and gone to sleep together days ago, if you hadn’t been so dogged. But you’ll die just the same, or worse. You might just as well lie down now and give it up. You’ll never get to the top anyway.’
‘I’ll get there, if I leave everything but my bones behind,’ said Sam. ‘And I’ll carry Mr. Frodo up myself, if it breaks my back and heart. So stop arguing!’
