* The Elder Wand's Loyalty: The Elder Wand's power is tied to the concept of "ownership" through conquest, not by killing the previous owner. This means simply killing the previous owner doesn't automatically transfer ownership. It needs to be won in a duel.
* Snape's Motives: Snape was not fighting for ownership of the Elder Wand when he killed Dumbledore. He was fulfilling a desperate plan by Dumbledore to ensure Voldemort's downfall and Harry's safety.
* Dumbledore's Plan: Dumbledore knew the wand's allegiance to Draco Malfoy, who had disarmed him earlier. He needed Snape to be the one to disarm Draco, making Snape the wand's new master. This ensured that Voldemort, who had disarmed Snape in the past, would not be able to truly wield the wand's power.
* Harry's Ownership: Harry ultimately became the true master of the Elder Wand because he disarmed Draco Malfoy in the final battle, effectively winning the wand's allegiance.
Therefore, even though Snape killed Dumbledore, he didn't become the Elder Wand's owner. Dumbledore's plan was ultimately to put the wand in a position where Voldemort could not truly wield it, and Harry was the one who finally earned its loyalty.