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 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.
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.