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?
That is me just me thinking out aloud, rather than any claim with much thought to it, or substance.
Payoffs and attitudes in Star Wars with lightsabers can be a strange thing; Kenobi in the Prequels’ (“This weapon is your life” WTF?!), Rey’s vibes and the YounglingSlayer 3000 (in comparison to the Sith dagger that “horrible things have happened with this”!), and the underwhelming fete of the Darksaber in Mandalorian season 3 (after years of establishing and building up a significance and importance out of it).