*Spoilers ahoy!* *Long post ahoy!*
Why does Luke believe this? I think it's a simple answer: he doesn't have the information that the audience has about the rise of Darth Vader. Luke hasn't seen the prequel movies. Luke has no idea what it was that led Anakin to turn to the Dark Side, what mistakes the Jedi Order might have made, so after having his own protege do the same, it's natural for him to come to the conclusion that the Jedi religion is poisoned.
But we, the audience, do have the prequels, in their imperfect glory, so we can look at Anakin and at Kylo Ren, and we can make a comparison more accurate than the one that Luke made. And indeed, The Last Jedi sets us up for exactly that. The Last Jedi is all about the differences between Anakin and Kylo Ren.
Anakin Skywalker was born into a harsh existence. Raised as a slave on a distant and brutal desert planet along with his single mother, he has a lot of trauma to deal with already by the time Qui-Gonn Jinn and Obi-Wan Kenobi come skipping into the picture. He's made the best of it, proving himself to be a gifted engineer as well as a strong Force user. But when Qui-Gonn frees him from slavery and offers to mentor him as a future Jedi, Anakin jumps at the chance. Still, joining the Jedi means leaving the one person he's always had on his side, his mother--and not just leaving her, but leaving her in the same tragic situation he's escaping, slavery on Tatooine. Though he chooses to leave, it's not without a share of heartbreak, and Anakin expresses his determination to come back and free not only his mother, but all the slaves in the Outer Rim.
This problem comes to its inevitable head in Anakin Skywalker. Obi-Wan does choose to mentor Anakin in memory of Qui-Gonn, but the other Jedi continue to treat Anakin with suspicion and distrust. Instead of recognizing the impact that his past has on his mental health and appropriately treating it, they stigmatize his struggles and try to force him to forget what he's been through. They teach him to swallow the trauma, never recognizing how much damage this can do. I don't know if I can say that Anakin has PTSD at this point, but he definitely is in need of a good counselor to help him work through what he's experienced. But that's not "the Jedi way," and so Anakin becomes emotionally dependent on the only three people who treat him well--Obi-Wan Kenobi, Chancellor Palpatine, and Padme Amidala. And then his mother, the one other person in the universe who he trusts, dies in front of him, beaten and bruised.
As such, it's only natural that Anakin would become frantic when he learns, though his Force powers, that Padme is going to die. Anakin doesn't just love Padme as his wife and the future mother of his children; he's dependent on her. And he's losing all of the people he trusts: now that Anakin's a full Jedi, Obi-Wan isn't around as much for him to turn to. Additionally, Anakin can't fully trust Obi-Wan: if he turns out to be more like the other Jedi than Anakin hopes, Obi-Wan could both hurt Anakin with his lack of approval and get Anakin kicked out of the Jedi Order. Meanwhile, the Jedi are telling Anakin not to trust Palpatine. With all that happening, how could Anakin be anything but desperate to save Padme?
While Darth Vader is indeed a villain, and while his actions cannot be justified, it's easy to understand how he got to this point--and why he would still have it in him to turn back to the Light. But Kylo Ren's life is a different story entirely.
Ben Solo is raised free and healthy, with both parents, receiving all the benefits of the new universe the Rebellion has created. Yes, his parents' relationship may have been a bit unstable, although I think it's more likely that the fractures we see in their relationship in The Force Awakens were caused by Ben's turn to the Dark Side, not the other way around. All evidence points to Leia and Han being good parents. And yes, Ben had a lot of heavy expectations on his shoulders, but so do many of us. Nothing in Ben's early life indicates the kind of trauma Anakin was dealing with. In fact, when you compare Ben's early life to that of every other character in the Star Wars universe, it's clear that he is the Star Wars definition of a privileged white boy.
Unfortunately, like many privileged white boys in our society, Ben becomes prideful. Rather than seeing his privilege as the result of the Light, he begins to think of himself as being somehow better than the others around him. And while his family sees him as a potentially incredible future Jedi, that isn't enough for him. They want him to protect others, to protect freedom, and that doesn't support his feeling of entitlement. So when Snoke comes along, Ben's eager to meet someone who did validate his feelings of self-importance. That's when Ben Solo becomes Kylo Ren--and it isn't as the result of years-long trauma, like with Anakin. It's the result of pure, simple pride, and it happens before Luke even trains Ben.
Feeling that his teaching wasn't quite reaching Kylo, Luke looked into Kylo's mind and saw very clearly that he had already turned to the Dark Side, and how so. In that moment, Luke's instinctive response was to destroy the monster that lay before him, so he drew his lightsaber. But that moment lasted only a split second, and then Luke was absolutely horrified with himself. (Some have said that Luke would never do that, even for a split second--I would like to point out that Luke is not a perfect man. He's not Jesus Christ, even if he does follow the archetypal path. Rewatch how he absolutely whales on Darth Vader before regaining control of his temper in Return of the Jedi, and I think you'll see what I mean. Any good person, when faced with evil, reacts with horror and, yes, at least a little anger--including Jesus Christ, who be throwing tables, my friends.)
Still, Luke is sufficiently affected by the result of his mistake, and he exiles himself so he can take a moment to think about what happened. Once he gets over the initial shock of what he has done, Luke must inevitably realize that his one mistake doesn't explain Kylo. After all, terrible as it was, by the time Luke made that mistake, Kylo had already turned. So Luke compares what little he knew of Anakin to what happened (or in fact, didn't happen) with Kylo, and comes to the conclusion that the fault is in the Jedi teachings themselves. Something about being a Jedi corrupts people, and Luke failed by being prideful enough to believe that he could teach the Jedi ways appropriately. Therefore, it's best that Luke remain alone on the island until he dies, taking the Jedi with him. A little overdramatic, perhaps, but that is the Skywalker Way (TM).
Luke may never know the truth of what happened with Anakin. Unlike the rest of us, he can't make a fair comparison. But when Luke, after having "lost" Rey, decides to destroy the Jedi texts themselves, he receives a surprise--Yoda shows up in his Force ghost form. Now, Luke may not know the truth, but Yoda does. Yoda was there for Anakin's turning, and undoubtedly, he's had some time to think about the mistakes that the Jedi Order made. Luke expects Yoda to try and stop him from destroying the texts, but instead, Yoda sets the tree that holds the texts on fire via lightning bolt. (Of course, Yoda, unlike Luke, probably knows that Rey took the texts with her, but the message still stands.)
In his own quirky way, Yoda is telling Luke the truth that Luke previously failed to see--yes, the Jedi Order, the old ways, may have been wrong. It doesn't matter, ultimately. Those ways aren't Luke's ways, and they aren't the important part of what it means to be a Jedi. While some parts of the past are better left behind us, the lessons learned from those events are still important, and that is what it means to be a teacher: passing on both success and failure and leaving the next generation to choose what to do with that information. Some will choose wrongly. Others will go on to become even better than their teacher. Such is the way with Kylo and with Rey--one chose the Dark and the other has chosen the Light. Yoda goes on to tell Luke that while he cannot save Kylo Ren from the Dark, he can save Rey and, by extension, the Resistance--so he better get off his butt and do it.
And in doing so, Luke becomes not the last Jedi.
Anakin, after he became Darth Vader, remained under the thumb of the new Emperor Palpatine for the next twenty-three years. The Emperor had control over everything, and Darth Vader saw no way and no reason to stop that. Again, terribly wrong, but understandable when you think about how broken Anakin was. It isn't until Luke, Anakin's own son, reaches out to him and, for all that Darth Vader did, continues to believe in him, that Darth Vader realizes there's another, better path. It isn't just Jedi Order and Empire, and salvation is possible, even for him--and, truly, deeply, Darth Vader doesn't want Luke and all his goodness to die. So Darth Vader kills the Emperor to save his son, and in so doing, both frees and redeems himself.
Kylo's relationship with Snoke is very different from Anakin's relationship with Palpatine. While Palpatine expertly manipulated Anakin so that he became emotionally dependent on him and then kept him under his thumb up until the end of Return of the Jedi, Kylo is never dependent on Snoke, never sees him as a friend. Snoke is a means to an end, and while Kylo might respect him in some ways, he isn't controlled by Snoke the same way Anakin was controlled by Palpatine. Kylo is allowed to act in ridiculous, childish ways, is allowed to throw temper tantrums and destroy property all the time. And when Snoke finally calls Kylo out on his childishness by insulting his helmet, it seriously ticks Kylo off. Kylo comes to the conclusion that it's time to "let the past die--kill it, if you have to." But he's still conflicted about his decision to kill Snoke--up until the moment he learns that Snoke has been manipulating him by connecting him with Rey. The threads of Kylo's last vestiges of loyalty snap, and he decide to murder "his true enemy."
Essentially, the message of the original Star Wars movies is that "anyone can be redeemed," but the message of The Force Awakens and The Last Jedi is a vital caveat to that fantastic original theme: "but not everyone wants to be." Sometimes, people just choose to do the wrong things, and that is the case with Kylo. Kylo, even from the beginning, is given every opportunity to join and rejoin the Light, and instead, he repeatedly chooses the Dark. Someone like that cannot be saved. They don't want to be.
Is it possible that Kylo will get a redemption arc in the last movie of the new trilogy? Sure. Anything's possible. But I really hope he doesn't, not because I want Ben Solo to be evil, but because that message is so, so important. It's important to people in abusive relationships, it's important to parents whose kids have gone totally off the rails, and it's especially important to all of us today facing a climate that gave birth to the alt right and far too many mass shooters. All of us need to know that sometimes, people choose not to be redeemed, and when that happens, it's our responsibility to do whatever is necessary to protect ourselves and those around us. If Kylo gets redeemed, that message is undercut, and it would be not just tone deaf to the modern state of the world (something that Star Wars has never been before), but also honestly a disappointment.
For now, I'm happy to say that The Last Jedi is a great movie with a great message. It's not a perfect movie, (as CinemaSins says, "no movie is without sin"), but I stand by my case.