Idk, it doesn’t seem like that that much of a stretch for Vader’s reputation to grow in political/Inner Rim circles, while being completely irrelevant to the day to day on a backwater planet without any consistent Imperial presence.

Ben is specifically keeping out of The Fight and to himself too. I think it makes a lot of sense and works towards establishing just how much he’s isolated himself. If it’s contrived, I don’t mind it. It’s small detail that says so much more about the character than the alternative IMO

Obi-Wan realizing Anakin is alive is also one of the only emotional beats that would have made this show worth making anyway, do we really want that to have just happened off screen?