Mystery Man
Kenobi Writer Explains Why Obi-Wan Didn't Know Anakin Survived
Jody Harold, the author of #ObiWan Kenobi, explains why and how the Jedi master was ignorant that his former disciple survived their battle on Mustafar

Obi-Wan last saw his student on the planet Mustafar, where he was left burning alive and walked away, although he was not present when the Emperor recovered Vader's body. Many audience members assumed Obi-Wan was aware of Vader's survival because of the 19-year gap between Revenge of the Sith and A New Hope, but it appears now that he truly believed his former apprentice was dead, leaving many to wonder how Obi-Wan went so long without hearing about the powerful Sith.
Obi-Wan Kenobi series writer Joby Harold discusses how Obi-Wan spent so long without knowing his former pupil had not only survived but was still alive in an interview with The Wrap. While this was not in the writer's first proposal, he realized while they were writing that this was an area that might be pursued and asked Lucasfilm Story Group executive and Obi-Wan Kenobi creative consultant Pablo Hildago whether this would break any canon. This plot point, Harold said, provided McGregor with more depth in his portrayal. Harold stated, "
“That was actually not something I pitched originally. That was something I discovered along the way, and sort of had to confirm with Pablo [Hidalgo] and really think, ‘Hold on a second, what does he actually know? Does he know the moniker Vader? What would that mean? Can he associate the two? What was he cognizant of? How isolated is he? Where’s Vader at that time? Where’s his reputation and how well known is he?’ and all those pieces of the puzzle. The the great piece of storytelling you can use is getting to Ewan play the moment of realization that that which haunts him is still alive and what does that mean for him? There were many avenues that I could take him down, all of which hopefully are good opportunities to tease out story as we continue. But it all comes down to, is it viable within canon to play that card? Which it was. Which is great, because that allows you, at the end of Episode 2 and the beginning of Episode 3, it gets you so much that feels essential to the fundamental story, which is Obi Wan, Vader, that which haunts you, facing the past, everything that comes to fruition in 3 and beyond.”

Obi-Wan Kenobi does establish that he is shut off from much of the rest of the galaxy, living in exile on Tatooine. It's easy to believe he doesn't connect with people or learn much about what's going on in the galaxy, so the term Darth Vader is unlikely to have come up in conversation with him. This also aligns with recent additions to the Star Wars canon via numerous novels and comics, which established Darth Vader and the Sith Inquisitors as Imperial enforcers but apart from the main Imperial troops, explaining why Darth Vader gets orders from Grand Moff Tarkin in A New Hope.
The scenario is eerily similar to one in Star Wars: Rebels, in which Ahsoka Tano realizes that the terrible Sith Darth Vader is also her former master. The series now adds a new dimension to A New Hope, explaining why Obi-Wan Kenobi is not astonished but knows Anakin's new appearance. Obi-Wan Kenobi, like the explanation for the Death Star malfunction in Rogue One: A Star Wars Plot, is filling in aspects from the original Star Wars picture that may not have occurred to anyone else but assist to deepen and enhance the entire story.
Source: The Wrap