Question: According to the poem:

> 
> Three Rings for the Elven-kings under the sky,  
> 
>  Seven for the Dwarf-lords in their halls of stone,  
> 
>  Nine for Mortal Men doomed to die,  
> 
>  One for the Dark Lord on his dark throne  
> 
>  In the Land of Mordor where the Shadows lie.  
> 
>  One Ring to rule them all. One Ring to find them,  
> 
>  One Ring to bring them all and in the darkness bind them  
> 
>  In the Land of Mordor where the Shadows lie.
> 
> 
> 

This implies that the One Ring has power over the other rings and those that bear them. I'm not fully knowledgeable about the state of the other rings in the setting of the Lord of the Rings trilogy but I'm under the impression that nine rings were worn by the Nazgûl were already under the power of Sauron. The rings given to the Dwarves were lost, or Sauron had recovered them. The rings owned by the elves could have been destroyed or safely hidden so to avoid being used by anyone.

Why was it imperative that the One Ring be destroyed if its control over the other ring bearers would be negligible?

Answer: 
The control that the One Ring has over the others is significant but that's not the real reason why it must be destroyed.

If we look at *Of the Rings of Power and the Third Age*, in *The silmarillion*, we'll see the reason why the One Ring was made and the reason why it must be destroyed:

> 
> And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency; and Sauron forged it in the Mountain of Fire in the Land of Shadow. And while he wore the One Ring he could perceive all the things that were done by means of the lesser rings, and he could see and govern the very thoughts of those that wore them.
> 
> 
> 

A key part of this is the statement that **much of the strength and will of Sauron passed into that One Ring**.

From there it's obvious: destroy the One Ring and you also destroy that part of Sauron's power that he put into it. Sauron as a result is greatly diminished, and is no longer able to act as a material threat to the world.
It is likely that Sauron's purpose in creating the Ruling Ring was indeed complete dominion over the Elves through their rings, but by the time of LOTR the Ring means something very different to Sauron and the Free Peoples. In addition to controlling the other Rings, The Ruling Ring focuses and multiplies the willpower of its master. Armies will flock to his banner. Wielding the Ring does not make one invincible, but it massively enhances one's military strength.

Even without the Ring, Sauron has gathered conventional military strength to extend control over Middle Earth, but he must fight many hard battles to achieve his end. Recovering the Ring would aid his campaign, but more importantly it would prevent the only thing that he believes can stop him, which is another person claiming the Ring and using it to build up their own military.

For the Free Peoples though, this would simply be to replace domination by one Dark Lord with another. While the Ring is in existence it is only a matter of time before somebody seizes and uses it, and this is the reason why it was imperative to destroy it. The control that the One Ring has over the Three was certainly not negligible – the Elves defences are dependent on them. But the reasons for destroying the ring go far deeper.