I agree that this keeps the film from being a perfect stand-alone, but there is a mitigating factor in that we do see a lightsaber fight so the lightsaber as an object is paid-off. The fact that it belongs to Obi-wan isn’t to troubling to me. Star Wars is a universe of unbounded promise, so giving Luke a weapon that he doesn’t use in a fight only fires the mind to imagine him using it in a future battle against Vader.
I would argue that the setup for Kenobi’s lightsaber happens in the Mos Eisley cantina, when Kenobi slices off that guy’s arm. This sets up that Kenobi is some kind of skilled warrior with an exotic weapon from a romantic age in the past. The payoff happens when Kenobi takes out his lightsaber again to fight Vader.
But Luke’s lightsaber is setup separately, and the setup connects the lightsaber to the idea that Luke’s father was a Jedi Knight who specifically wanted Luke to inherit it. (Thus, the lightsaber serves as a physical manifestation of the “hero’s call” - calling Luke away from his mundane life to adventure.) But there’s never any pay off for this setup in A New Hope. It’s kind of like setting up Excalibur in the King Arthur legends, but then just forgetting about it.
A case could also be made the Kenobi himself is the physical manifestation of the “Heroes’ call”? Which would relegate the lightsaber to being a tool of that?
I remember reading somewhere (I forgot where I read this - could be in Rinzler or could be Secret History of Star Wars) that the scene where Ben gives Luke the lightsaber was included mostly as setup for later when Ben uses his own lightsaber in the Mos Eisley Cantina. The Cantina scene where Ben dismembers someone goes back to one of the earliest drafts of Star Wars. However, Lucas realized that if the Cantina scene was the first time the audience saw a lightsaber, nobody would understand what even happened. So the scene with Luke playing around with his father’s lightsaber was added to introduce the audience to the concept of lightsabers.
So you’re probably correct. But I can just imagine an alternate reality where Star Wars 1977 is the only Star Wars movie to exist, where an audience member might easily think (especially upon rewatching it) “Hey… wait, what’s the point of Luke getting a lightsaber? He never uses it.” Usually in most of these fantasy movies, if the hero is given a magical weapon, the hero will use that weapon at some point (usually in the film’s climax) to overcome some obstacle or defeat the bad guy. (The weird 80s fantasy movie Krull comes to mind as a typical example of this happening, and of course in Tolkien there are multiple occasions where some magical object/weapon is setup for a later payoff.)