How wouldn't it "make sense" in ANH? So you're saying that just because we don't see Obi Wan physically picking up the saber in ROTS, that the audience can't connect the dots themselves and assume he's recovered (Jedi) Anakin's discarded saber off camera at some point?

Are you Lucas? lol

Regardless, seeing Obi Wan picking up the lightsaber after cutting off Anakin's legs and leaving him to burn to death just makes you think "what a lying bastard" in ANH when he says "your father wanted you to have this"

It has to go!

At least this way, he's not giving Luke a saber he used as Darth Vader that was used to murder children.

And as for the hilt in the battle matching the one Sidious gives him...who on earth has a quick enough eye to spot it unless they know what they're looking for...plus Vader's saber didn't match between ANH, ESB and ROTJ anyway. In fact, the ROTJ one was completely different and made from an entirely different camera flash because they lost the ANH/ESB prop.