Whose Wand Is It Anyway?

alrighty . . . 

Jumping into the Deathly Hallows . . . one in particular -- the Elder Wand.

A friend and I argued about the Elder Wand and whether or not the wand exists given the confusion of Elder Wand Mastery in Harry Potter and the Deathly Hallows. 
     She ended up winning because I couldn't figure out how to phrase my views. I've been thinking about it ever since and I've finally found my words. 

So here we go . . . 

At the time, it sounded like I didn't believe the wand existed at all. Instead, the right of Mastery is passed down to people and there is no official, tangible Elder Wand in existence . . .
     but that's not what I was trying to say. 

I acknowledge that the wand exists and truly is the most powerful wand in the world BUT you don't need the wand to be master.   

As we all know . . . 

Draco never touched the Elder Wand in his life, but the Elder Wand aligned itself  to him anyway. Therefore, the power of the Elder Wand was given to Draco. 

In that same fashion, Harry stole Draco's unremarkable wand and earned Mastery of the wand at Malfoy Manor.

Again . . . this is stuff we already know.

The confusing part is what happens from this point. I am probably weird and odd in my thinking, but here it goes anyway. 
     Draco doesn't touch the Elder Wand, but he has mastered it, therefore  Draco's wand is the new Elder Wand.
     In taking Draco's wand, Harry becomes master of the Elder Wand.
     Also, the actual Elder Wand never works for Voldemort. It's power comes solely from Voldemort, a great wizard. So, wouldn't Voldemort need Draco's wand to be master?

I know I'm crazy -- In the book, Harry puts the wand back with Dumbledore. In the movie, Harry breaks the wand in half. 
      Meaning that there MUST be a tangible wand, but the mastery doesn't necessary rely purely on ownership of the wand.

Does that make any sense at all?  Or have I honestly lost my mind?

Thoughts?