He couldn't bring himself to kill Darth Vader, however, because he continues to associate him with the Anakin he trained as a boy. This is also the exact reason why Obi-Wan didn't kill Anakin on Mustafar despite the opportunity a decade earlier in Star Wars: Episode III - Revenge of the Sith.
Obi-Wan knew it was only a matter of time. So, he made the difficult decision to use his learnings and become one with the force, rather than let Vader decapitate him.
Originally Answered: Why did Obi-Wan Kenobi let Darth Vader defeat him? Because he wanted Luke the get the Hell out of there, and he knew Luke wasn't going to leave his mentor behind if there was a chance to save him.
Obi-Wan also let Vader go because Vader wanted his old master to kill him. Sometimes being left alive is more of a punishment than death. But now Obi-Wan could walk away knowing there was no hope for saving Anakin — until Luke finds a way.
To conclude, Obi-Wan let himself die because he thought he would become more powerful and more useful to Luke as a Force Ghost. Aside from this, you could also look at the fact that it provided the distraction that allowed Luke, Leia, Han and Chewbacca to escape the Death Star.
Yes he did. And the saddest thing is that there was nothing he could do about it. He was broken. The first case is shortly after ROTS when Vader was making his Sith lightsaber.
Darth Vader showed no emotion at all when he killed Obi-Wan Kenobi. If you watch the scene carefully, you will see Kenobi pausing. He was letting Vader kill him. Whether Vader knew this at the time, or discovered it later, is an open question.
"Goodbye Old Friend, May The Force Be With You"
Anakin admits to Obi-Wan that he has been arrogant and unappreciative of his training, and apologizes to his master. Obi-Wan assures Anakin that he is proud of him, telling him that he has become a "far greater Jedi than I could ever hope to be."
Despite everything they have been through, the Obi-Wan Kenobi show has firmly established that the Jedi still deeply cares for his former Padawan and remembers him as the great ally he's had for many years. Seeing Anakin as Darth Vader understandably breaks Obi-Wan's heart.
Theory 1: Leia Doesn't Remember Ben Is Obi-Wan
Throughout her encounters with Kenobi in the series, he refers to himself as Ben. The most straightforward explanation would be that Leia knows the person who rescued her is Ben, and never connected the dots that Ben is actually Obi-Wan.
Obi-Wan Kenobi, during his confrontation with Vader on Mustafar, stated that he believed Anakin was indeed the Chosen One, but that his destiny had been altered, and the prophecy left unfulfilled. Anakin fulfills the prophecy by killing Palpatine. Skywalker's sacrifice brings balance and redemption.
Move aside, Anakin Skywalker. Master Yoda has risen above all others to claim the top spot as the most powerful Jedi of all time and the true chosen one according to IGN's audience. He won pretty handily as well, as he was victorious in 89,756 of his 95,243 battles and had a win percentage of 94.2%.
During the iconic opening moments of Star Wars: A New Hope, Darth Vader has a tense confrontation with Princess Leia, an adversary who, unknown to him, is actually his daughter. Despite participating in Leia's interrogation, the Sith Lord doesn't uncover his familial connection with the rebellious royal.
Why does Obi-Wan Kenobi age like that? Because he's not real, and is a fictional character played by different people. You see, Obi-Wan Kenobi is played by Alec Guinness in the original trilogy. Guinness was 62 when he filmed A New Hope, and he died in 2000.
As for why he didn't just kill him, It's because Obi Wan was a Jedi. Anakin was literally an Unarmed down opponent, and Obi Wan lived by a code you don't kill an unarmed down opponent because that would lead down the a path to the darkside, See Luke not killing Darth Vader in Return of the Jedi.
The reason Obi-Wan failed to reach Anakin was that he blamed himself for Anakin's fall to the dark side. To him, Vader had become a nightmare that reminded him of his past. He needed Vader to be redeemed not for Anakin's sake, but for his own.
Yes, when Darth Vader turned back to Anakin Skywalker and sacrifice himself to save his son, he dies and Obi-Wan uses the force to connect with Anakin's spirit. In that time Anakin apologizes to Obi-Wan, desperately asking him to forgive him for everything he has done to him and everybody over the years.
He slays Sidious and returns to Obi-Wan, begging for atonement by execution, only to be met with true forgiveness instead. But just as this fantasy Obi-Wan refuses to strike his former friend down, Vader refuses to turn his back on the dark.
The Jedi Council's Rules on Attachments
The choice to leave his mother at a young age continues to haunt Anakin throughout his life. He is plagued by nightmares of her and visions of her fate. When he eventually saves her from a group of Tuskens on Tatooine, his blind anger at her dying state consumes him.
Apparently, he had borderline personality disorder. As he came of age, Skywalker showed problems with impulsivity, violent outburst, illusions of invincibility and crises of identity, all of which are in line with borderline personality disorder diagnosis, the researchers concluded.
In the material around the film from the Legends timeline, it's established that by the end of Star Wars: Episode III - Revenge of the Sith, Kenobi was one of the characters who knew Anakin's Sith name was Darth Vader. However, what he didn't know was that he was still alive after Mustafar.
In the duel, Anakin lost both of his legs, his one remaining organic arm, and suffered (possibly up to) fourth-degree burns across what was left of his body that were severe enough to damage his internal organs.
Over the years since the Youngling massacre, and even sometime before, Darth Vader has regretted killing the Younglings, recognizing that it was the most terrible thing he had ever done.
Star Wars Confirms Darth Vader's Love for Padme Amidala Was Immortal.