I am wondering why Odin didn’t have Stormbreaker since it is a ‘King’s Weapon’, or was the magical staff/spear that he carried around in the Thor movies more powerful than Stormbreaker?
I can think of a few reasons really for why he didn't want it.
It was symbolic
Gungnir was symbolic of the campaign that Odin led to Conquer the Nine Realms because it was his weapon during it. He's then used it to maintain the peace and to lead afterwards. This was a symbol of power for Asgard so it makes sense he'd then keep using it even just for his largely ceremonial role nowadays as Thor is doing a lot of the fighting.
It appears to be tradition to use your "own" weapon
During Thor's ceremony Odin says the below which indicates somewhat that keeping the one weapon during your rule is somewhat of a tradition. Thor would never have gotten Stormbreaker if Mjolnir wasn't destroyed so that backs up the theory that they keep their one weapon usually.
Odin: And though the day has come for a new King to wield his own weapon -- that duty remains the same. Thor Odinson, my heir, my first-born. So long entrusted with this mighty hammer, Mjolnir. Forged in the heart of a dying star, from the sacred metal of Uru. Only one may lift it. Only one is worthy. Who wields this hammer commands the lightning and the storm. Its power has no equal -- as a weapon, to destroy, or as a tool, to build. It is a fit companion for a King.
He didn't need it
Gungnir is clearly a powerful weapon, we get a glimpse of Bor using it at the beginning of Thor: The Dark World where he defeats the Dark Elves. It's power is then seen several times afterwards with Loki using it a bit too. He managed to defend Asgard and the Nine Realms with it so he clearly didn't really need a new one.
Odin: Gungnir. Its aim is true, its power strong. With it I have defended Asgard and the lives of the innocent across the Nine Realms since the time of the Great Beginning.
Gungnir can even open the Bifrost as we see in Thor when Odin banishes Thor. He might have to use it more as a key than being able to summon the Bifrost like Stormbreaker can but Odin didn't really venture out anymore. He didn't have much need for summoning the Bifrost on the fly so the upgrade wasn't worth it. Presumably though Gungnir could summon the Bifrost like we saw Heimdall do with the broken Hofund at the start of Avengers: Infinity War it just takes a bit more.
Your familiarity with a weapon matters
It may be obvious to some but familiarity with a specific weapon matters a lot and even more so when switching between types of weapons. Gungnir is a staff/spear with a ranged attack whereas Stormbreaker is a big old axe. They require vastly different fighting techniques which Odin might not have been comfortable with so wouldn't have wanted it.
We see this somewhat in the duel Thor has with Hela whilst wielding Gungnir, he is easily bested. In part this is due to him not awakening his power and Hela being super strong but he was also unfamiliar with Gungnir being mainly a hammer user until then. He didn't use Gungnir effectively because he was unfamiliar with it.
Stormbreaker wasn't created yet
Lastly, and it's quite important, Stormbreaker wasn't created yet and the theory on the weapon wasn't even completed. Eitri states "In theory, it could even summon the Bifrost." meaning it was probably still in the prototype stage. If that wasn't completed and the weapon wasn't even built it makes sense why Odin wouldn't use it.