“Owning” (curiously rarely “renting”) the bandstand implies buying up not sharing and a permanence for the duration of the gig at the very least.
Because at its most advanced level jazz is about collective improvisation whatever dialect of the language is selected and that collective process of creation based on a whole lot of preparation or not is happening like right now when people play.
At its worst owning the bandstand is a power struggle. It might be a blast for the player “owning the bandstand” but for the crowd that can translate into domination and subordination. An hour-and-a-half of ego followed by a 20-minute break and then more ego until close is a tad dull.
Power struggles achieve little certainly not great music making. Do you want the band to be a vehicle for the main player or not? That process might occur anyway in the direction of travel of the music.
Does anyone own the bandstand? Yes. But not who you think because players are at the mercy of what they are creating that has crossed to the ear of the beholder even as they participate in its creation. The genie is well and truly out of that bottle.