I wouldn't say it's an "overwhelming despair" but for me it's just a "general sadness" that looms over the story.

The first 3 books (in publication order) all start with the children in a state of mostly boredom in the real world, and then Narnia comes along to offer a sense of escapism and adventure.

Silver Chair by comparison opens with the children in a state of mild peril and distress. Getting bullied at school is a miserable experience. Getting into Narnia is less of a "call to adventure" as a relief.

Plus then when they get there - Caspian being an old man is sad, his wife being dead is sad, Rilian having been kidnapped is sad. 

Is not overwhelmingly bleak no, and yes there are still moments of levity in there, but perhaps the BBC series doesn't quite capture that sense of general sadness?

For what it's worth though, the BBC Narnia series represents such an intrinsic part of my childhood, that they are pretty much immune from criticism for me... Despite how objectively crummy the production values are in places, I can't do anything but love them.

The beginning of The Silver Chair is somewhat gloomy, but it is not overwhelmingly pessimistic. The book begins with a desire to leave our world for something better, e.g. Jill wanting to escape the bullies.  There is hope offered to those who want something better.  The story has a dark tone at the beginning which changes to a more positive experience when the children reach Narnia, although there is still some sadness with the disappearance of Rilian and the lingering illness and later death of King Caspian.

I wouldn't call it "overwhelming sadness" either, but the start of The Silver Chair IS very bleak — I was re-reading it the other day after listening to the Talking Beasts podcast. The previous three books don't have any direct threat to the main characters before they find themselves in Narnia, whereas SC opens in a school that is emphatically an absolutely awful place. (And Lewis was writing from experience — the overriding impression I got from Surprised by Joy was not how compelling his journey from atheism to Christianity was, but how ghastly nearly all of his real-life schools were.)

As someone who has experienced long-term bullying at school and indeed in a former workplace, I wouldn't shrug off Jill and Eustace's situation as just mildly upsetting. The bullies at this school get away with their actions because they know “the right sort of things to say to the Head”, who lets them off every time. For their victims, there's no recourse to the authorities who should be putting a stop to the bullying. And don't forget, they're in a boarding school. The bullies are lying in wait all the time — there's no going home at the end of the day where they can't get you. (I always think, thank goodness I left school just a few years before cyberbullying became a thing.) So Jill and Eustace's entry into Narnia isn't just a relief — it's an escape from what is, to them, pretty much a prison. Lewis brings that across very strongly in the opening chapter: “all the hopelessness of Experiment House (it was a thirteen-week term and there were still eleven weeks to come)”. It's a pity the BBC adaptation doesn't manage to convey that oppressive atmosphere nearly as well. I hope any future adaptation will pick up on it a lot more!

Actually, I find The Silver Chair the second-bleakest of all the stories (after the first two-thirds of The Last Battle, naturally) — there's a dreariness throughout most of the entire book. After the awful school, the exciting mysteriousness of the wood beyond the door is soon broken by the squabble on the cliff and the horror of Eustace's fall. And then Aslan, when Jill meets him, is far more stern and forbidding than in most of his previous appearances in the other books. A reader who hadn't read the earlier books wouldn't know this is the Lion who gave his life to save a traitor, who rose from the dead and romped with Susan and Lucy, who brought stone statues back to life, who brought liberation and joy back into Narnia in Caspian's time (with another romp!), who undragoned Eustace, who was there at the end of the voyage to the edge of the world and revealed that he is in our world too, under another name... (And if you’re reading in chronological rather than publication order, you’ve also seen the sheer awesomeness of Aslan creating a new world through his song, and the poignancy of him granting Digory the means to save his mother’s life, but that hadn’t yet been written at the time Lewis was writing this book.) If you've only seen him in this story, this talking lion comes across as a deeply unsettling and rather scary character, and there's almost nothing in the rest of the story — until the very end — to show his tender side or to give some real perspective on just why Jill ought to trust him.

Not to mention that the rest of the plot has very little levity. Caspian is old and dying, his wife has been killed, his son has disappeared, and the journey to find the Prince (with the gloriously miserable Puddleglum, who DOES provide some laughs, ironically enough!) takes us through a harsh and cold landscape, through an encounter with supposedly "gentle" yet sinister giants, and then into the darkness of the Underland... It's a great story and a great adventure, but there’s far less beauty and brightness in it than in most of the other books. Even in LB, the mounting horror as Narnia is destroyed gets turned around, wiped away, by the joy and wonder of discovering Aslan’s country. We do get a tiny glimpse of that at the end of SC, when Caspian is brought back to life — the promise that death is not the end and that one day Jill and Eustace too “will have come to stay” here in Aslan’s country with him — but that’s about it. 

In nearly all the other books, there’s at least one sustained scene of utterly radiant wonder — the creation of Narnia, the resurrection of Aslan, his return in PC, the last stage of the voyage in VDT, and (again) the finale of LB. I just don’t find that in SC. It’s still a fantastic story and there’s a lot in it I love (especially Puddleglum and his magnificent speech as he breaks the Green Witch’s spell), but I do find it almost the darkest part of the whole Narnia saga and I’m always a bit surprised at how many people here rate it as their absolute favourite of the books!!