harry potter - Why Did Voldemort Kill Snape?

I know that the answer to that question is to obtain the Elder Wand, but I think that approach doesn't really make sense.

We know that death is not necessary for the Elder Wand to change hands (Grindelwald stole it from Gregorovitch, and Albus obtained by beating Grindelwald in a non fatal duel) which means we know Voldemort didn't have to kill Snape to obtain the wand, that was just one way of doing it.

Snape was a powerful wizard(maybe one of the strongest of the death eaters) and had proven himself more useful than almost any Death Eater(providing years of information on Dumbledore and later killing him), Voldemort probably didn't want to kill him unless it was absolutely necessary.

Couldn't there have been some other way to get the wand from Severus? Like challenging Snape to a non-fatal duel for ownership of the wand, or perhaps even giving Severus the wand and then stealing it, Grindelwald style?

Answer

1. 
   

   Because Voldemort is sure that killing is the only way.

   
   

   All the theories of winning its allegience that you listed in the question ("We know that death is not necessary") never even occurred to Voldemort, whose solution to pretty much any problem is "Avada Kedavra"

   
   

   
   

   The Elder Wand cannot serve me properly, Severus, because I am not its true master. The Elder Wand belongs to the wizard who killed its last owner. You killed Albus Dumbledore. While you live, Severus, the Elder Wand cannot truly be mine.”

   
   

   
   
   


2. 
   

   Because Snape became less useful at that point, so the downsides of killing him aren't there to balance the decision.

   
   

   
   

   —and it is doing so without your help,” said Voldemort in his high, clear voice. “Skilled wizard though you are, Severus, I do not think you will make much difference now. We are almost there... almost.

   
   

   
   


3. 
   

   Please note that it was literally, the only solution that occured to him, and he honestly didn't just do it because he was tired of Snape.

   
   

   While he may have been play-acting for Snape in their dialogue later in the scene, he obviously was NOT play-acting when talking to Nagini earlier:

   
   
   

   
   

   “Snape. Now. I need him. There is a—service—I require from him. Go.”
   Frightened, stumbling a little through the gloom, Lucius left the room. Voldemort continued to stand there, twirling the wand between his fingers, staring at it.
   “It is the only way, Nagini,” he whispered, and he looked around, and there was the great thick snake, now suspended in midair, twisting gracefully within the enchanted, protected space he had made for her, a starry, transparent sphere somewhere between a glittering cage and a tank.

   
   

   
   

All quotes are Deathly Hallows, Chapter 32, "The Elder Wand"