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,  
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>  One for the Dark Lord on his dark throne  
> 
>  In the Land of Mordor where the Shadows lie.  
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>  One Ring to rule them all. One Ring to find them,  
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>  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.
The One Ring would:

* Solidify the control of the Ringwraiths and empower them toward greater evil than they were already capable of.
* Prevent the use of the 3 Elven rings because as long as Sauron had the One Ring, no one would consider using the 3 in any way for fear of being controlled, manipulated and eventually taken over as a powerful remote under Sauron's control.
* It would return a great deal of Sauron's capacity and power to him, since he bound much of his fea (spiritual energies) within the One Ring.

Destroying it ensured he would never have access to that power again and would go from being one of the strongest powers in Middle Earth to fading away to nothing.